REESE  LIBRARY 


UNIVERSITY  OF  CALIFORNIA. 

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PRINCIPLES  OF  MECHANISM. 

A  TREATISE  OX 

THE  MODIFICATION  OP  MOTION 


BY   MEANS   OP   THE 


ELEMENTARY  COMBINATIONS  OF  MECHANISM, 
OR  OF  THE  PARTS  OF  MACHINES. 


FOR  USE  IN  COLLEGE  CLASSES, 

BY  MECHANICAL  ENGINEERS, 

ETC.,  ETC. 


BY 

STILLMAN  W.  ROBINSON,  C.E.,  D.So., 

Mechanical  Engineer  and  Expert  for  the  Wire  Grip  Fastening  Co.  ;  Vice- President 
and  Mechanical  Engineer  to  the  Grip  Machinery  Co.  ;  till  recently  Professor 
of  Mechanical  Engineering  in  the  Ohio  State  University ;  Member 
Am.  Soc.  Mechanical  Engineers ;  Member  Am.  Soc.  Civil 
Engineers ;  Fellow  of  the  American  Association  for 
the  Advancement  of  Science  ;  Member  Am.  Soc. 
Naval  Architects  and  Marine  Engineers  ; 
and  Member  Soc.  for  Promo- 
tion of  Engineering 
Education. 


FIRST   EDITION. 

FIRST  THOUSAND. 


NEW  YORK: 

JOHN  WILEY  &  SONS. 

LONDON  :  CHAPMAN  &  HALL,  LiMirm 

1896. 


BY 

S.  W.  ROBINSON, 


ROBERT  DRUMMOND,    ELECTROTYPER  AND   PRINTER,   NEW  YORK. 


PREFACE. 


THIS  work  aims  to  treat  the  whole  subject  of  MECHANISM  in 
such  systematic  and  comprehensive  way  that  by  its  aid  any  machine, 
however  elaborate,  may  be  analyzed  into  its  elementary  combina- 
tions, and  the  character  of  their  motions  determined. 

In  the  classification,  the  System  of  Prof.  Robert  Willis  has  been 
followed  in  the  main,  as  serving  best  the  present  purpose;  and 
largely  his  names  and  terms  as  well. 

The  work  contains  the  substance  of  lectures  given  in  my  classes 
during  the  past  twenty-seven  years,  with  such  additions  and 
amplifications  as  might  come  by  reason  of  a  somewhat  extended 
study  of  the  subject  not  only,  but  as  brought  out  in  connection 
with  an  aptitude  for  the  thinking  out  of  inventions  involving  more 
or  less  novel  and  varied  forms  and  combinations. 

Besides  Willis,  some  of  the  authors  whose  works  have  been 
referred  to  with  profit  may  be  named:  Prof.  C.  W.  MacCord; 
Geo.  B.  Grant;  J.  W.  M.  Rankine;  F.  Reuleaux;  J.  B.  Belanger; 
Oh.  Laboulave. 

Some  of  the  topics  to  which  special  attention  is  invited,  as  em- 
bracing either  entirely  new  solutions  of  important  questions,  or 
previously  unpublished  discussions  and  extension  of  inquiry  con- 
cerning them,  are: 

Log-spiral    Multilobes    as    derived  from   one    spiral;    also 

Proportional  Sectors. 
Easements  to  Angular  Pitch  Lines. 
Transformed  Wheels. 
General    Solution   of   Non-circular  Wheels,    External    and 

Internal,  for  the  case  of  Given  Laws  of  Motion. 
Similar  and  other  Wheels  from  Auxiliary  Sectors,  Plane  and 
Bevel. 

ill 


iv  PREFACE. 

Special  Bevel  Non-circular  Wheels  laid  out  on  the  Normal 

Sphere. 
General  Solution  for  Bevel  Non-circular  Wheels  for  Stated 

Laws  of  Motion. 
General   Solution  for  Skew-bevel  Non-circular  Wheels  for 

Stated  Laws  of  Motion. 
Practical  Rolling  of  Pitch  Lines  and  Engagement  of  Teeth, 

in  one  Pair  of  Non-circular  Wheels. 
Intermittent  and  Alternate   Motions   for   Moderate   or   for 

High  Speeds;  Circular  and  Non-circular. 
Internal  or  Annular  Non-circular  Wheels. 
Teeth  for  Skew-bevel  Non-circular  Wheels. 
"  Blocking,"  and  Steepest  Gear  Teeth. 
Interference  of  Involute  Teeth  of  Annular  Wheels. 
Epicycloidal  Engine  and  Accessories  for  Machine-made  Teeth. 
Full  Discussion  of  Olivier  Spiraloids;  Interference,  etc. 
Cam  Construction  by  Co-ordinates. 
Form  of  Roller  for  Cams. 

Solution  for  Cams  with  "  Flat  Foot  "  Follower. 
Cam  of  Constant  Breadth  and  Given  Law  of  Motion. 
Easements  for  Cams. 

Solutions  for  Varied  Velocity-ratio  in  Belt  Gearing. 
Non-circular  Pulleys  for  Continuous  Motion  by  Law. 
Solution  for  Cone  Pulleys. 
Rolling  Curve  Equivalent  for  Link-work  in  General ;  Plane 

and  Bevel. 

Gabs  and  Pins  for  Link- work  in  General ;  Plane  and  Bevel. 
Velocity-ratio  in  Bevel  and  Skew-bevel  Link-work. 
A  General  Crank  Coupling   connecting  Shafts  in  Various 

Planes  and  Angles. 

Practical  Forms  of  Parts  for  Bevel  and  Skew-bevel  Link- 
work. 

Varied  Step  Ratchet  Movement. 
Face  Ratchets  and  Clicks. 

The  aim  of  the  work  is  not  so  much  to  present  a  history  of 
Mechanism f  as  to  treat  upon  the  principles  which  underlie  the 
various  modes  of  modification  of  motion  as  due  to 'the  form  and 
connection  of  parts,  and  thus  to  enable  the  inventor  and  designer 
of  machines  to  at  once  solve  any  problem  of  motion  of  an  elemen- 
tary combination  in  a  particular  case. 


PREFACE.  V 

The  treatment  has  been  mainly  by  graphics  instead  of  by  analysis, 
for  three  reasons:  1st,  because  the  draftsman's  outfit  is  usually  at 
hand  when  mechanism  problems  arise;  3d,  because  these  problems 
usually  do  not  require  the  precision  of  analysis,  the  graphic  method 
serving  for  problematic  work  as  well  as  for  delineation;  3d,  because 
analysis,  though  possible  in  a  few  of  the  simpler  problems,  becomes 
difficult  and  often  impossible  with  very  slight  variations  of  condi- 
tions, while  by  the  graphic  method  all  cases,  whether  of  simple  or 
complex  statement,  are  solved  with  nearly  equal  facility.  Hence 
analysis  has  been  employed  here  only  to  establish  principles  the 
application  of  which  subsequently  might  be  made  by  the  graphic 
method. 

S.  W.  ROBINSON. 
COLUMBUS,  O.,  Sept.  26,  1896. 


CONTENTS. 


INTRODUCTION. 

OBJECT  OF  PRINCIPLES  OF  MECHANISM  Page 

How  Studied.  Machine  Defined.  Frame  ;  and  Trains  of  Mecha- 
nism. Machine  Parts  Classified.  Elementary  Combination  of 
Mechanism.  Motions,  how  Studied,  1.  Synoptical  Table  of  Ele- 
mentary Combinations  of  Mechanism,  2.  Velocity  ;  Constant  and 
Varied ;  Angular.  Velocity-ratio.  Period.  Cycle,  3.  Revolution. 
Distinction  into  Driver  and  Follower.  Directional  Relation,  4. 


PART   I. 
TRANSMISSION  OF  MOTION  BY  ROLLING   CONTACT. 

CHAPTER  I. 

ROLLING  CONTACT  IN  GENERAL Page 

Line  of  Contact.  Axes.  Plane  of  Axes.  Point  of  Contact. 
Line  of  Centers.  True  Rolling  Contact.  Velocity-ratio  in  Circular 
Rolling  Contact.  No  Slip  of  Surfaces  in  Contact,  5.  Velocity- 
ratio  of  Non  circular  Arcs,  6.  Variable  Velocity-ratio,  7. 

CIRCULAR  WHEELS Page 

Axes  Parallel.  Friction  Wheels.  Contact  between  the  Axes, '7. 
Contact  Outside  the  Axes.  Axes  Meeting,  8.  Parallelogram  for 
Locating  Line  of  Contact.  Rolling  Cones.  Angle  between  Axes, 
Particular  Case,  9.  Axes  Crossing  without  Meeting.  Character  of 
Contact  between  the  Surfaces.  Form  of  the  Rolling  Surfaces. 
Location  of  Line  of  Contact  in  Plan,  10.  Location  of  Line  of  Con- 
tact in  Elevation.  Hyperboloidal  Form  of  Surfaces.  Longitudinal 
Slip  of  Surfaces.  Error  in  Early  History  of  Case,  11.  Graphic 
Construction  of  Skew  Bevels.  Proof  of  Construction,  12.  Not 
Practical  for  Friction  Wheels.  Longitudinal  Slip  Determined,  13. 

CIRCULAR  INTERMITTENT  MOTIONS Page 

For  Axes  Parallel.  Locking  of  Driven  Wheel  when  Idle. 
Avoidance  of  Friction  Clamps.  Positive  Locks  Classified,  13.  Work- 

vii 


Till  CONTENTS. 

ing  Drawings  of  Various  Circular  Intermittent  Motions,  and  Illus- 
trations of  Actual  Constructions  of  Same,  14-18. 

CHAPTER  II. 

SPECIAL  NON-CIRCULAR  WHEELS , Page    19 

Complete  or  Incomplete.  Theory  of  Pitch  Lines.  Axes  Fixed. 
Three  Cases :  Axes  Parallel,  Axes  Meeting,  and  Axes  Crossing 
without  Meeting.  Notation.  True  Rolling  Contact.  Point  of 
Contact  on  Line  of  Centers,  19.  Radii  in  Pairs  ;  their  Sum  a  Con- 
stant. Mutually  Rolled  Arcs  equal  Each  Other.  Elementary 
Sectors.  Five  Special  Cases.  Equal  Log-spirals.  Will  Roll  Mutu- 
ally and  Correctly,  20.  To  Construct  the  Log-spiral  Graphically. 
Geometric  Series  of  Radii.  Spiral  Passing  Two  Given  Points,  21. 
Length  of  Log-spiral  Arc.  Application  of  Log-spiral  Wheels.  Sec- 
toral Wheels.  Log- spiral  Levers,  22.  Weighing  Scales.  Wire 
Cutter.  Wipers.  Complete  Wheels,  23.  Symmetrical  and  Uusym- 
metrical  Log-spiral  Wheels  :  By  Aid  of  Two  Spirals  ;  and  by  Angles 
and  Radii,  24.  Multilobed  Wheels  :  by  Reduction  of  Angles  :  by 
Assumed  Angles  and  Limiting  Radii,  25.  Interchangeable  Multi- 
lobed Log-spiral  Wheels  :  as  derived  from  One  Spiral  with  Lobes 
Symmetrical ,  as  derived  from  Two  Spirals  with  Lobes  Uusymmet- 
rical,  26.  Interchangeable  Log-spiral  Lobed  Wheels.  One,  and 
Three  Lobed  Wheels,  as  determined  from  Proportional  Sectors, 
27,  28.  Case  of  Infinite  Number  of  Lobes,  29.  Easement  Curves. 
Equal  and  Similar  Ellipses.  Proof,  29.  Range  of  Velocity-ratio, 

30.  Example   from   Centennial   of   1876  of  Elliptic   Gear  Wheels, 

31.  Interchangeable    Multilobed    Elliptic    Wheels,    Holditcli,    32. 
Working  Drawings  and  Photo-process  Copy.     Wheel  with  Infinite 
Number  of  Lobes,  33.      Equal  and  Similar  Parabolas  as  Rolling 
Wheels.     Equal  Hyperbolas,  35,  36.      Transformed  Wheels.     Three 
Rules,  37.     Changes  Characteristic  of  Application  of  Each  Rule, 
38.      Interchangeable    Multilobes  by  Transformation,    39.      Trans- 
formed   Elliptic    Wheels,    40.       Transformed    Parabolic    Wheels. 
Wheels  of  Combined  Transformed  Sectors,  41,  42. 

CHAPTER    III. 

NON-CIRCULAR  WHEELS  IN  GENERAL Page    44 

Three  Cases  :  One  Wheel  Given  to  Find  its  Mate  ;  Laws  of  Motion 
Given  to  Find  the  Wheels ;  Similar  and  Other  Wheels  from 
Auxiliary  Sectors.  Complete  or  Incomplete.  Given  One  Wheel, 
to  Find  its  Mate,  44.  Repeated  Trials.  Graphic  Rules  for  Equiva- 
lence of  Lines  and  Arcs,  45.  To  Draw  a  Tangent  to  any  Curve,  46. 
Application  of  Above  Rules  to  Case  of  One  Wheel  Given  to  Find 
its  Mate,  47.  Examples  of  Above.  Laws  of  Motion  Given  to  Find 
the  Wheels,  48,  49.  Proof  of  Above,  50,  51.  Photo-process  Copies 
from  Examples  of  Above,  52,  53.  Solutions  of  Some  Practical 


CONTENTS.  IX 

Problems,  in  Laws  of  Motion  Given  to  Find  the  Wheels.  The 
Shaping-machine  Problem,  54,  55.  A  Bobbin-winding  Problem  ; 
Solution  for  Conical  Bobbin,  56,  57.  Motion  of  Driver  ii  Variable. 
Case  of  Multilobed  Wheels.  Similar  and  Other  Wheels  from 
Auxiliary  Sectors,  58.  Similar  and  Equal  Multilobed  Wheels. 
Dissimilar  Multilobed  Wheels,  59.  Multilobes  of  Unequal  Number 
of  Lobes,  60,  61. 

CHAPTER  IV. 

.SPECIAL  BEVEL  NON-CIRCULAR  WHEELS Page    62 

Normal  Spheres.  Spherical  Equiangular  Spiral.  Oue-lobed 
Spherical  Equiangular  Wheels,  62.  Drawing  on  Spherical  Blank 
for  Pattern,  63.  Two-lobed  and  Multilobed  Bevel  Non -circular 
Wheels.  Interchangeable  Bevel  Non-circular  Wheels,  64.  Ellip- 
tic Bevel  Wheels,  65.  Multilobed  Elliptic  Bevel  Wheels.  Parabolic 
and  Hyperbolic  Bevel  Wheels.  Transformed  Bevel  Wheels,  66. 
Similar  Bevel  Multilobes.  Dissimilar  Interchangeable  Bevel  Multi- 
lobes.  Partially  Interchangeable  Bevel  Multilobes.  Bevel  Non-cir- 
cular Wheels  of  Combined  Sectors,  67.  Interchangeable  Multilobes 
with  Uusymmetrical  Lobes.  Velocity  ratio  in  Bevel  Non-circular 
Wheels,  68. 

CHAPTER   V. 

BEVEL  NON-CIRCULAR  WHEELS  IN  GENERAL Page    69 

Laws  of  Motion.  Plane  Wheels  Turned  into  Bevels,  69.  Diagram 
of  Plane  Wheels  Possessing  Correct  Laws  of  Motion.  Auxiliary 
Diagram  giving  Spherical  Radii  from  Plane-wheel  Radii,  70. 
Spherical  Blank  for  Gear  Pattern.  One  or  Several  Lobed,  71.  Proof 
of  Above,  Example,  72.  Laying  Out  Non-circular  Wheels  Direct  on 
the  Normal  Sphere.  Case  of  One  Wheel  Given  to  Find  its  Mate. 
Also,  Laws  of  Motion  Given  to  Find  the  Wheels.  Similar  Wheels. 
Lobed  Wheels,  etc.,  73-4. 

CHAPTER  VI. 

SKEW-BEVEL  NON-CIRCULAR  WHEELS Page    75 

Illustrative  Hyperboloids.  Pitch  Surfaces  of  Form  of  Non- 
circular  Hyperboloids,  75.  First  Find  the  Plane  Non-circular  Wheels 
of  Correct  Law,  which  Turn  into  Skew  Bevels,  76.  Spherical  Blank 
to  be  Worked  into  a  Gear  Pattern.  Mode  of  Procedure,  77-79. 
Proof  of  Above,  80. 

CHAPTER  VII. 

^ON-CIRCULAR  WHEELS  FOR  INTERMITTENT  MOTIONS Page    81 

Classification.  Cases  of  Axes  Parallel,  Meeting,  or  Crossing 
without  Meeting.  Easement  Spurs  and  Locking  Arcs  of  Greater 
Radius  than  Rolling  Arcs.  Easement,  Locking  Arcs,  and  Teeth  in 


CONTENTS. 

One  Plane,  with  Radius  of  Locking  Arcs  Less  than  that  of  the  Rolling 
Arc,  81.  Laying  Out  of  the  Pitch  Curves.  Photo-process  Copies  of 
Actual  Bevel  Wheels.  Alternate  Motion  Gearing,  82.  Limited  Alter- 
nate Motions.  With  Velocity-ratio  Constant,  or  Varying,  83.  Un- 
limited Alternate  Motions.  Mangle  Rack,  84.  Mangle  Wheel,  85. 
Axes  Meeting,  86. 


PART   II. 

TRANSMISSION  OF  MOTION  BY  SLIDING   CONTACT. 
CHAPTER  VIII. 

SLIDING  CONTACT  IN  GENERAL Page    8T 

Velocity-ratio  in  Sliding  Contact,  87.  Proof  of  Velocity-ratio,  88. 
Sliding  and  Rolling  Curves  with  Common  Law  of  Motion.  One 
Sliding  Curve  Given,  Find  a  Mate,  With  Law  of  Motion  that  of  a 
Given  Pair  of  Rolling  Wheels,  89.  Teeth  of  Gear  Wheels,  91. 
Tracing  of  Sliding  Curves.  Odontoids,  Centroids.  Describing 
Templet,  92.  Names,  Terms,  and  Rules  for  Gear  Teeth,  93.  Pro- 
portions for  Gear  Teeth.  Circumferential  and  Diametral  Pitch,  94. 

CHAPTER  IX. 

TOOTH  CURVES  FOR  NON-CIRCULAR  GEARING.     GENERAL  CASE  . . .  Page    95- 

Axes  Parallel.  Templets,  95.  Use  of  Templets  in  Describing 
Tooth  Curves,  96.  Requirements  of  Toolh  Curves,  97.  The  Tooth 
Profile.  Position  of  Tracing  Point  on  Describing  Curve,  98.  The 
Trachoidal  Tooth  Curve,  99.  Form  and  Size  of  Describing  Curve, 

100.  Individually  Constructed  Teeth.     Non-circular  Involute  Gears, 

101.  Conjugate  Teeth,  102.     Limited  Inclination  of  Tooth  Curves, 
103.       Hooking  Teeth,    104.       Nearest  Approach   of  Tooth  Curve 
Toward  Radius,   105.      Practical    Limit    of    Eccentricity   of    Pitch 
Line,  106.     Eccentricity  as  Affected   by  the  Tooth  Profile,  and  the 
Addendum,  107.      "Blocking"  of  the  Gears,  and    as  Affected  by 
Length  of  Tooth,  108.      Substitution    of  Pitch    Line    Rolling    for 
Teeth  in   Extreme   Eccentricity.     Link  Substitute  for   Teeth,    109. 
Examples  and  Photo-process   Copies  of   Actual   Gears,    110,    111. 
Internal  Non-circular  Gears,  111. 

CHAPTER  X. 

TEETH  OF  BEVEL  AND  SKEW-BEVEL  NON-CIRCULAK  WHEELS Page  112i 

General  Case  of  Axes  Intersecting.  Describing  Curve  a  Cone,  112. 
Tracing  of  Tooth  Curve  Direct  on  Sphere.  Tredgold's  Approximate 
Method.  Involute  Teeth,  113.  General  Case  of  Axes  Crossing 
without  Meeting.  Teeth  Twisted.  Correct  Exact  Construction  not 
Possible  174, 


CONTENTS.  Xk 


CHAPTER   XI. 

NON-CIRCULAR  INTERMITTENT  MOTIONS Page  llfr 

Teeth  Spurs  and  Segments.  Teeth  as  in  Non-circular  Wheels. 
Engaging  and  Disengaging  Spurs.  Backlash,  116.  Spurs  of 
Rolling,  or  of  Sliding  Curve  Form.  As  Adapted  for  High  Speed,  or 
Slow,  118.  Rolling  Spurs  often  Impracticable,  119.  Rolling  Spurs 
require  Excessive  Arc  of  Motion.  Bevel  and  Skew-bevel  Wheel 
Spurs,  120.  Solid  Engaging  and  Disengaging  Segments.  Forms 
such  as  not  to  "Block,"  121.  Undercut  so  as  to  Relieve  the  Shock, 
122.  Counting  Wheels.  Alternate  Motions.  Limited  in  Move- 
ment, 123.  Skips  and  Derangements.  Spurs  Employed,  124. 
Solid  Segments  for  Reversing  Motion.  Best  Form  of  Wheel  for 
High  Speed,  125.  Axes  Meeting.  Normal  Sphere,  126.  Use  of 
Motion  Templets.  Unlimited  Alternate  Motions.  Mangle  Wheel 
and  Rack.  Non-circular  Form.  Velocity- ratio,  127. 

CHAPTER  XII. 

TEETH  OF  CIRCULAR  GEARING Page  128- 

Axes  Parallel.  Epicycloidal,  Involute,  and  Coujugatel  Gearing. 
Epicycloidal  Gearing,  128.  Some  Peculiar  Properties  of  Epicycloids 
and  Hypocycloids.  White's  Parallel  Motion,  129.  Seven  Styles  of 
Teeth,  viz.:  Flanks  Radial,  Concave,  Convex  ;  Interchangeable  Sets; 
Pin  Gearing ;  Rack  and  Pinion  ;  Annular  Wheels.  Flanks  Radial. 
The  Face  and  Flank  Generated  by  Describing  Circles,  130.  Flanks 
Concave.  Size  of  Describing  Circle.  Flanks  Convex,  132.  Inter- 
changeable Sets.  "Change  Wheels."  Describing  Circle,  133.  Pin 
Gearing,  134,  135.  Inside  Pin  Gearing.  Pinion  of  Two  Teeth,  136. 
Rack  and  Pinion.  Wheels  of  Least  Crowding,  137,  138.  Rack  and 
Pinion.  Flanks  Radial.  Flanks  Concave,  139.  Annular  Wheels. 
Peculiar  Case  of  Interference  and  an  Extra  Contact.  Wheels  in 
Sets,  140,  141.  Pin  Annular  Gearing.  Involute  Gearing.  Curves 
Described  by  Log-spirals  Rolled  on  Pitch  Line,  142.  Practical  Mode 
of  Drawing  the  Teeth,  143.  Rack  and  Pinion.  Annular  Wheel  and 
Pinion.  Interference  of  Annular  Wheel  Teeth,  144.  Wheel  of 
Several  Styles  of  Teeth.  Conjugate  Gearing.  Flanks  Parallel 
Straight  Lines,  145.  Flanks  Convergent  Straight  Lines.  Flanks 
Circle  Arcs,  146. 

CHAPTER  XIII. 

PRACTICAL  CONSIDERATIONS Page  14$ 

Addenda  and  Clearance.  Rule  for  Circumferential  Pitch.  Rule 
for  Diametral  Pitch,  148.  To  Strengthen  the  Teeth.  Possible 
Clearing  Curve.  Path  of  Contact :  Determined,  Assumed,  Limited, 
149,  150.  Acting  Part  of  Plank.  Line  of  Action.  Obliquity  of 
Line  of  Action,  151.  The  "  Blocking  "  Tendency.  Uusyrnmetrical 


ill  CONTENTS. 

Teeth.  Practical  Construction  of  Tooth  Curves,  152.  Tooth  Tem- 
plet, 153.  Radius  Rod  for  Tooth  Templet.  To  Draw  Involute 
Teeth,  154.  Approximate  Teeth  in  Practice,  155.  The  Templet 
Odontograph.  As  a  Ready-made  Tooth  Templet.  Radius  Rod,  156, 
157.  The  Willis  Odontograph,  gives  Center  for  Circle  Arc  Tooth 
Curves,  158-160.  The  Three  Point,  or  Grant's  Odontograph,  160, 
161.  Co-ordinating  the  Tooth  Profile,  162.  Advantages  of  Each  of 
Above  Instruments,  163. 

CHAPTER  XIV. 

MACHINE-MADE  TEETH Page  164 

Teeth  with  no  "Hand  and  Eye"  Process.  Epicycloidal  Engine 
to  Form  Epicycloidal  Curve,  164.  Grinding  the  Tool  to  the  Curve, 
165.  Forming  the  Gear  Cutting  Tool,  166.  The  Brown  &  Sharpe 
Involute  Cutters  ;  also  their  Epicycloidal  Cutters,  by  Machinery  of 
Pratt  &  Whitney  Co.,  167.  Sang's  Theory  of  Conjugating  the 
Teeth.  The  Swasey  Engine  with  Split  Cutter,  168.  The  Grant's 
Engine  with  Solid  Worm  Cutter,  170  Form  of  Cutter  Teeth,  171. 
The  Gleason's,  Corliss',  and  Bilgram's  Gear  Planing  Machine. 
Stepped  and  Spiral  Spur  Gearing,  172,  173. 

CHAPTER    XV. 

CIRCULAR  BEVEL  GEARING Page  174 

Correct  and  Approximate  Solutions.  The  First  by  Use  of  Cones 
with  Bases  formed  to  the  Normal  Spheres  ;  and  the  Second  by  Use 
of  Normal  Cones  of  Tredgold,  174.  Complete  Drawing  by  Tred- 
gold's  Method,  175.  Spiral  Bevel- wheel, Teeth,  176. 

CHAPTER  XVI. 

TEETH  FOR  CIRCULAR  SKEW-BEVEL  GEAR  WHEELS Page  177 

Approximate  Construction.  A  Practical,  Theoretically  Correct 
Solution  not  Known.  Approximate  Epicycloidal  Form.  Same 
Arbitrarily  Dressed  or  "  Doctored,"  177.  Tredgold's  Method. 
Applied  on  Normal  Cones,  178.  Amount  of  Error  Illustrated,  180. 
Exact  Solution  in  Olivier  Spiraloids,  181.  Olivier  Spiraloid  Explained. 
Intel-changeability  of  Spiraloids,  182.  Interference  of  Spiraloid 
Teeth.  Nature  of  Contact  of  Spiraloid  Teeth,  183.  Results  of  an 
Example.  Principle  Demonstrated.  Flat  Faces  of  Teeth  when 
Remote  from  Gorge  Circles,  184.  Certain  Forms  Approximating 
the  Olivier  Gears.  Olivier  Worm  and  Gear,  187.  Hiudley  Gears. 
Skew-pin  Gearing.  Intermittent  Motions,  188,  189. 

CHAPTER  XVII. 

CIRCULAR  ALTERNATE  MOTIONS Page  190 

Limited  Alternate  Motions.  Solid  Engaging  and  Disengaging 
Parts,  190.  With  Attached  Engaging  and  Disengaging  Parts.  The 
Mangle  Wheel.  Unlimited  Alternate  Motions,  191. 


CONTENTS.  X11I 


CHAPTER  XVIII. 

CAM  MOVEMENTS Page  192 

Cams  in  General.  Friction.  Backlash.  Solution  by  Co-ordinates, 
192.  By  Intersections.  Use  of  Templets.  Velocity-ratio,  193,  194. 
Continuously  Revolving  Cam,  195.  Cylindric  Cam,  with  Straight 
Path  for  Follower,  196.  Same  with  Curved  Path  of  Follower. 
Conical  Cam,  197.  Spherical  Cam,  198.  Cam  Plate.  Flat-footed 
Follower,  with  Specific  Law  of  Motion  of  Follower.  Uniform 
Motion,  199.  Law  of  Motion  that  of  Crank  and  Pitman.  Law  that 
of  a  Falling  Body,  200.  201.  Tarrying  Points.  Uniform  Recipro- 
cating Motion.  The  Heddle  Cam,  202.  Easements  on  Cams.  Ar- 
bitrary Easements,  and  those  Confined  to  Assigned  Sectors,  203. 
Case  of  Flat-footed  Follower,  205.  Cams  of  Constant  Diameter,  206. 
Cams  of  Constant  Breadth,  207.  Cams  with  Several  Followers.  The 
Effect  of  Two  Followers,  208.  Four-motion  Cam.  Illustration  from 
Example,  210.  Duangle  Cam.  Return  of  Follower  by  Gravity,  211. 
Positive  Return :  Three  Ways,  212.  To  Relieve  Friction,  213. 
Modification  of  Cam  to  Suit  Form  of  Follower,  or  Roller,  214,  215. 
Best  Form  of  Roller.  Action  of  Roller  upon  its  Pin  or  Shoulder. 
Roller  for  Conic  Cam.  For  Spherical  Cam,  216,  218. 

CHAPTER  XIX. 

INVERSE  CAMS  AND  COUPLINGS  ..... Page  2191 

The  Pin  and  Slot.  Form  of  Slot.  Velocity-ratio,  219.  Oldham's 
Coupling,  with  Shafts  in  Parallel.  220.  Shafts  not  Parallel,  221. 
Peculiar  Coupling.  Oldham's  Three-disk  Coupling,  223,  224. 

CHAPTER  XX. 

ESCAPEMENTS . .  Page  226- 

Power  Escapements,  225.  Anchor  Escapement,  226.  Pin-wheel 
Escapement,  228.  Gravity  Escapement,  229.  Cylinder  Escapements, 
231.  Lever  Escapement,  232.  Duplex  Escapement,  233.  Chronom- 
eter Escapement,  234. 

PART   III. 
BELT  GEARING. 

CHAPTER  XXI. 

TRANSMISSION  OP  MOTION  BY  BELTS  AND  PULLEYS.     ROPE,  STRAP,  OR 
CHAIN    OVER     SECTORAL    AND    COMPLETE     PULLEYS.      MOTION 

LIMITED  OR  CONTINUOUS.    VELOCITY-RATIO  VARYING Page  23ft 

Velocity -ratio,  236.  Line  of  Action.  Non-circiilar  and  Circular 
Pulleys.  Law  of  Perpendiculars  given  to  Find  the  Wheels,  237. 


:xiv  CONTENTS. 

Equalizer  of  Gas-meter  Proven  A  Draw-bridge  Equalizer,  239. 
Barrel  and  Fusee,  240.  Non-circular  Pulley  for  Rifling  Machine, 
241.  Example  of  Treadle  Movement,  243. 

CHAPTER  XXII. 

CIRCULAR  PULLEYS Page  244 

Continuous  Belt.  Velocity-ratio.  "Slip,"  244.  Retaining  Belt 
on  Pulley.  Pulley  with  High  Center,  245.  Crossed  Belt.  Quarter- 
twist  Belt.  Guide  Pulley,  246.  Any  Position  of  Pulleys.  Cone 
Pulleys.  Problem  of  Cone  and  Counter  Cone.  Geometric  Series  of 
Speeds,  247,  251.  Rope  Transmission :  Two  Systems.  Rope  Belting, 
Short  and  Long  Stretch,  251-53.  For  Haulage  Lines.  Compensating 
System.  Chain  and  Sprocket  Wheel.  Teeth  of  Sprocket  Wheels. 
Practical  Application  of  Chains,  254-57. 

PART   IV. 
LINK-WORK. 

CHAPTER  XXIII. 

HODS,  LEVERS,  BARS,  ETC.     THE  GENERAL  CASE Page  258 

Velocity-ratio,  258.  Peculiar  Features  of  Link- work  Mechanism. 
Lightest-running  Mechanism.  Indexible  Law  of  Motion.  Axes 
Parallel,  259.  Examples  :  1.  Needle-bar  Motion ;  2.  Corliss'  Valve 
Gear ;  3.  Driven  Piece  Half  the  Time  nearly  Quiet ;  4.  Small  Move- 
ment in  a  Given  Time,  260,  261.  Path  and  Velocity  of  Points. 
Sliding  Blocks  and  Links,  261-63. 

CHAPTER  XXIV. 

.A  ROLLING  NON-CIRCULAR  WHEEL  EQUIVALENT  FOR  EVERY  PIECE  OF 

LINK-WORK.     GABS  AND  PINS Page  264 

Examples  showing  Several  Rolling  Curves,  264.  Curves  and 
their  Linkages  shown  Separately,  265,  266.  Rolling  Curve  Equiva- 
lent of  Crank  and  Pitman.  Two  Cranks  and  Drag  Link.  The 
Sylvester  Kite,  267,  268.  Equal  Cranks  in  Opposite  Motion.  Hyper- 
bolas. Elliptic  Wheels,  269-71.  Parabolic  Wheels.  Unsymmetri- 
cal  Link-work  and  Rolling  Wheels,  272,  273.  Dead  Points  in  Link- 
work.  "  Gab  and  Pin."  Path  of  Gab  and  Pin.  Gab  and  Pin  in 
Practice,  274-77.  Multiplying  Motion  by  Link-work,  278. 

CHAPTER  XXV. 

OONIC  LINK- WORK .   Page  279 

Axes  All  Meet  in  a  Point.  Cranks  Equal  or  not.  Velocity-ratio, 
279.  Rolling-wheel  Equivalent  of  Conic  Link-work.  Dead  Center, 
and  Gab  and  Pin  in  Conic  Link-work.  Examples  of  Peculiar  Move- 


CONTENTS.  XV 

ments,  281.  Possible  Valve  Motion.  Hooke's  Joint  or  Coupling, 
282-85.  Almond's  Coupling,  286.  Crank  Coupling,  287.  Reuleaux 
Coupling,  288. 

CHAPTER   XXVI. 

LINK- WORK  WITH  AXES  CROSSING  WITHOUT  MEETING Page  289 

Ball  and  Socket  or  Sliding  Joints.  Velocity- ratio,  289.  The 
Willis  Joint  System,  290.  Shafts  Connected  by  Cranks  and  Angle 
Blocks,  Three  Examples,  291,  292.  Ratchet  and  Click  Movements. 
Running  Ratchet.  Varied  Step  Movement,  292,  293.  Reversible 
Running  Ratchet.  Reversible  Varied  Rate  Running  Ratchet.  Con- 
tinuous Running  Ratchet.  Forms  of  Teeth  and  Click,  294,  295. 
Friction  Ratchets.  Fluted  Bearing.  Continuous  Friction  Ratchet. 
Wire  Feed  Ratchet.  Ruuniug  face  Ratchet  and  Click,  296,  297. 


PART   V. 
REDUPLICA  TION. 

CHAPTER  XXVII. 

PULLEYS  OR  SHEAVES  AND  ROPES Page  298 

Velocity-ratio.  Parallel  Ropes,  298.  Ropes  not  Parallel.  Veloc- 
ity ratio.  Hay  Unloader,  299.  Hydraulic  Elevators.  Weston 
Differential  Block.  Velocity- ratio,  300. 


PRINCIPLES  OF  MECHANISM. 


INTRODUCTION. 

IN  working  out  the  design,  drawings,  and  specifications  for  a 
machine,  the  form,  strength,  and  motion  of  the  various  parts 
must  be  determined,  the  last  being  the  object  of  the  Principles 
of  Mechanism. 

In  Principles  of  Mechanism  we  find  the  application  to  machines, 
of  the  principles  of  Kinematics,  or  Cinematics,  the  elementary 
combinations  of  mechanism  of  which  machines,  being  studied 
separately. 

A  Machine  may  be  defined  as  a  combination  of  fixed  and  mov- 
ing parts  or  devices,  so  disposed  and  connected  as  to  transmit  or 
modify  force  and  motion  for  securing  some  useful  result. 

The  fixed  parts  constitute  the  frame  or  supports  for  the  moving' 
parts. 

The  moving  parts  constitute  a  train  or  trains  of  mechanism. 

A  train  of  mechanism  may  be  primary  or  secondary;  the  former 
being  supported  directly  by  the  frame,  and  the  latter  by  other 
moving  parts. 

All  the  moving  parts  of  machines  may  be  regarded  as  mechani- 
cal devices  and  classified  as  follows: 

1st.  Revolving  shafts. 

2d.  Revolving  wheels  or  cams,  with  or  without  teeth. 

3d.  Rods  or  bars  with  reciprocating  or  vibratory  motion,  or 
both. 

4th.  Flexible  connectors  depending  on  friction. 

5th.  Flexible  connectors  independent  of  friction. 

6th.   A  column  of  fluid  in  a  pipe. 

Trains  of  mechanism  consist  of  combinations  of  the  above* 
devices,  the  least  number  securing  a  modification  of  force  or  motion 
in  a  given  case,  being  an  elementary  combination. 

The  study  of  the  motions  of  a  machine  is  usually  pursued  by 


PRINCIPLES   OF   MECHANISM. 


taking  up  separately  the  elementary  combinations  of  mechanism 
composing  the  machine. 

Professor  Robert  Willis  of  Cambridge,  England,  was  the  first  to 
present  a  thorough,  systematic,  and  comprehensive  table  of  the 
elementary  combinations  of  all  mechanism.  In  our  study  of 
mechanism  to  include  all  kinds  and  varieties  without  omissions, 
we  can  do  no  better  than  to  follow  this  table,  as  below : 

SYNOPTICAL    TABLE    OF    THE    ELEMENTARY    COMBINATIONS 
OF    MECHANISM.     (WILLIS.) 


Mode  of 
Transmission  of 
Motion. 

Directional  Relation  Constant. 

Directional  Relation 
Changing  Periodically. 

Velocity-ratio 
Constant. 

Velocity-ratio 
Varying. 

Velocity-ratio 
Constant  or  Varying. 

By   rolling    con- 
tact. 

Rolling  cylin- 
ders, cones,  and 
hyperboloids; 
pitch-circles  of 
circular     gear  - 
wheels  and  sec- 
tors. 

Pitch-lines  of  non- 
circular  gear- 
ing, complete  or 
sectoral. 

Pitch  -lines  for 
mangle  -  wheels 
and  mangle- 
racks;  limited  al- 
ternate motions. 

By   sliding    con- 
tact. 

Tooth  -  curves; 
segmental  cams; 
screws;  worm 
gearing. 

Tooth-curves  for 
noncirculargear- 
ing  ;  earns  ;  pin 
and  slotted  le- 
ver ;  irregular 
worm  gearing  ; 
stop  motions. 

Cams  in  general; 
pin  and  slotted 
lever;  double 
screw  ;  swash- 
plate  ;  escape- 
ments. 

By  wrapping  con- 
nectors, or  belt 
gearing. 

Band  or  belt  and 
pulleys  ;    chain 
and      sprocket- 
wheel. 

Cam-shaped  pul- 
ley and  belt; 
fusee  and  chain 
or  chord. 

Cam  -  shaped  pul- 
ley and  lever,  or 
tightener  ;  trea- 
dle motion. 

By  link-work. 

Equal  cranks  with 
link;  lever  with 
proportion  al 
links;  bell-crank 
and  links. 

Equal  cranks  with 
link  ;  unequal 
cranks  with 
link  ;  Hooke's 
universal  joint. 

Crank  or  eccentric 
and  pi  tma  n  ; 
ratchet-  wheels 
and  clicks;  un- 
equal cranks  and 
link. 

By  reduplication. 

Cord  and  pulley; 
pulley  -  blocks 
with    parallel 
ropes  or  chains. 

Pulleys,  with  rope 
or  chain  not  par- 
allel. 

NAMES  AND  TERMS. 

In  the  study  of  mechanism  certain  terms  are  used,  some  of 
which  are  defined  below : 

Velocity. — Time  is  required  for  a  point  to  move  a  distance  along 
a  line  or  path.  For  uniform  motion  the  distance  moved  over  per 
unit  of  time  is  the  velocity,  usually  considered  as  feet  per  second. 

Thus  velocity  is  the  rate  of  motion,  or  movement. 

For  the  case  of  uniform  motion,  any  distance  passed  over  in  a 
corresponding  given  time,  divided  by  that  time,  gives  the  velocity. 

When  the  velocity  is  not  constant,  the  space  passed  over  in  a 
very  short  time  is  divided  by  that  short  time ;  as,  for  instance 
0.3  ft.  in  0.1  sec.,  when  the  velocity  will  be 

^  =  3  ft.  per  sec., 

which  is  the  velocity  or  rate  of  moving  at  the  instant  considered. 

In  mathematical  calculations  the  space  and  time  are  often 
reduced  to  infinitesimals  for  extreme  exactness,  but  this  is  rarely 
necessary  in  the  study  of  mechanism. 

Angular  Velocity. — Velocity  may  here  be  distinguished  as  linear 
and  angular  ;  the  former  being  the  rate  of  motion  along  any  line, 
while  the  latter  is  the  rate  of  motion  of  a  point  at  a  distance  unity 
from  a  center  of  angular  motion.  Either  may  be  constant  or 
variable.  If  variable,  the  rate  is  to  be  expressed  for  an  instant 
by  taking  the  ratio  of  an  element  of  space  to  the  corresponding 
element  of  time. 

Velocity-ratio. — This  is  the  ratio  of  two  angular  or  two  linear 
velocities;  or,  in  some  cases,  of  an  angular  and  a  linear  velocity. 

Period. — Moving  parts  of  machines  continue  in  the  repetition 
of  certain  definite  complete  movements. 

The  time  for  one  of  these  complete  movements  may  be  called  a 
period,  though  this  term  is  but  little  used  in  mechanism. 

Cycle. — The  time  for  several  moving  pieces,  considered  collec- 
tively, to  return  to  their  given  initial  positions  may  be  called  a 

3 


4  PRINCIPLES   OF   MECHANISM. 

cycle,  though  this  term  is  of  comparatively  little  use  in  mechanical 
movements. 

Revolution. —A  revolution  is  to  be  considered  a  complete  turn, 
while  the  term  rotation  may  be  applied  to  any  portion  of  a  turn. 

Driver  and  Follower. — In  any  elementary  combination  of  mech- 
anism one  piece  always  drives  the  other,  the  one  therefore  being 
called  the  driver,  and  the  other  the  follower. 

Directional  Relation. — This  term  has  reference  to  the  relation 
of  the  directions  of  motion  of  driver  and  follower.  When  one 
never  reverses  its  direction  of  motion  unless  the  other  does  also,  the 
directional  relation  is  constant,  otherwise  it  is  said  to  be  variable. 


PART  I. 

TRANSMISSION    OF  MOTION  BY  ROLLING 
CONTACT. 


CHAPTER  I. 
KOLLING  CONTACT   IN  GENERAL. 

Rolling  Contact  in  elementary  combinations  of  mechanism  im- 
plies a  driver  which  turns  a  follower  by  rolling  against  it  without 
slipping,  the  driver  and  follower  having  axes  of  motion  which  are 
at  a  constant  distance  apart. 

The  Line  of  Contact  of  thick  pieces  is  always  a  straight  line  like 
that  of  the  contact  of  a  pair  of  parallel  straight  cylinders  tangent 
to  each  other. 

Sometimes  reference  is  had  to  the  plane  of  the  axes,  or  plane 
which  contains  both  axes. 

The  line  of  contact  of  thick  pieces  in  true  rolling  contact  will 
evidently  remain  in  the  plane  of  the  axes,  and  in  transverse  sections 
the  point  of  contact  or  of  tangency  of  the  rolling  curves  of  section 
must  likewise  remain  on  the  line  joining  the  centers  of  motion, 
called  line  of  centers. 

It  is  evident  that  in  true  rolling  contact  no  slipping  can  occur, 
and  that  arcs  which  have  rolled  correctly  in  mutual  contact  must 
be  equal  to  each  other. 

VELOCITY-RATIO   IN   ROLLING   CONTACT. 

First.  For  Circular  Arcs  and  Cylinders. — Suppose  Fig.  1  to  rep- 
resent a  transverse  section  of  a  pair  of  rolling  circular  cylinders 
with  axes  at  A  and  B.  The  line  of  centers  is  AB,  and  C  on  that 
line  is  the  point  of  contact.  Take  Cb  =  Ca,  and  draw  the  radii 
R  and  r.  Now  if  Ca  rolls  without  slipping  on  Cb,  a  and  b 

5 


PKINCIPLES   OF  MECHANISM. 


will  eventually  come  into  contact  at  C  on  the  line  of  centers,  when 

R  will  have  the  position  A  C,  and 
r  the  position  BC\  while  the 
axis  A  will  have  turned  through 
the  angle  CA  a,  and  the  axis  B 
through  the  angle  CBb.  The 
point  e  will  describe  the  arc  ed, 
and  the  point  g  the  arc  gf.  If 
FIG.  1.  these  movements  occur  in  one 

second  of  time  with  uniform  motion,  and  if  e  and  f  are  taken  at  a 

units  distance  from  A  and  B  respectively,  then 


ed  =  angular  velocity  of  A  =  V; 
gf  —  angular  velocity  of  B  —  v. 
Ae  \R\\de  :  Ca  ::  V :  Ca, 
Bg  :  r  \\  fg  :  Cb  ::  v  :  Cb  or  Ca; 
V  x  R  =  Ca  x  Ae  =  v  X  r  =  Cb  x  Bg. 


and 
Also 

and 


Whence        -==5  =  ^, 


snce 


Ca  =  Cb,    and     Ae  —  I  =  Bg, 


which  is  the  velocity  ratio  of  the  axes  A  and  B  for  motion  trans- 
mitted from  A  to  B  by  the  arc  Ca  rolling  on  the  arc  Cb,  and  equals 
the  inverse  ratio  of  the  radii  R  and  r. 

As  this  is  true  of  any  transverse  section  of  the  cylinders,  it  is 
true  of  the  entire  cylinders.  Hence  the  important  principle,  viz.: 
the  velocity-ratio  of  truly  rolling  cylinders  is  equal  to  the  inverse 
ratio  of  the  radii  of  those  cylinders. 

Second.  For  Non-circular  Arcs  and  Cylinders. — Let  Fig.  2 
represent  a  transverse  section  of  a  pair  of  correctly  rolling  non-cir- 
cular cylinders,  or  portions  of 
them,  with  A  and  B  as  axes. 

Then  AB  is  the  line  of  centers,  A 

""  GB 


C  the   point   of    contact,   a  Ac 

and  bBd   the  non-circular  sec- 

tors.     Draw  circular   arcs  eCg 

and  fCh,  and  the  velocities  V  FlG-  2- 

and  v  as  shown.     Then  V  is  common  to  a  Ac  and  eAg.    Also  v  is 

common  to  bBd  and  ff>h,  and  the  velocity-ratio  is  the  same  for 

the  non-circular  as  for  the  circular  sectors,  the  angles  to  the  sectors 

being  regarded  as  very  small. 


CIRCULAR    ROLLING    WH 


The  segments  of  the  line  of  centers  being  the  same  for  the  non- 
circular  as  for  the  circular  sectors,  we  have 


Velocity-ratio 


BO 

AC> 


that  is:  the  velocity-ratio  in  non-circular  wheels  at  any  instant  is 
equal  to  the  inverse  ratio  of  the  segments  of  the  line  of  centers. 

As  these  segments  are  continually  varying  in  non-circular 
wheels,  it  follows  that  the  velocity-ratio  for  such  wheels  is  also 
variable. 

CIRCULAR  ROLLING  WHEELS. 
DIRECTIONAL   RELATION    CONSTANT.      VELOCITY -RATIO    CONSTANT. 

Friction  Wheels  and  Pitch  Lines  for  Circular  Gearing. 

For  this  subject  the  wheels  are  circular,  with  velocity-ratio  con- 
stant; and  we  have  three  general  cases. 

I.  Axes  Parallel.  For  this  case  there  are  two  subdivisions. 
First :  Contact  between  the  axes  as  in  Fig.  3.  Second :  Contact 
outside  the  axes  as  in  Fig.  4. 


FIG.  3. 

These  friction  ivlieels,  or  pitch  lines,  as  the  case  may  be,  are  so- 
simple  in  the  theory  of  mechanism  that  no  further  treatment  seems 
necessary,  unless  the  following  graphic  problems  are  considered. 

First.  Contact  Between  the  Axes. — The  axes  being  at  a  fixed 
distance  apart  and  parallel,  let  them  be  represented  in  Fig.  5  by  the 
lines  at  A  and  B.  Take  the  velocity-ratio  as  being  given  with  the 
value  2/5.  Then 


8 


PRINCIPLES    OF   MECHANISM. 


As  the  sum  of  the  radii  equals  the  distance  between  the  axes, 

A then  if  we  take  r  —  2  inches  and  7^  =  5 

inches,  and  add,  we  have  7  inches.  If 
this  be  less  than  the  distance  between 
the  axes,  double  each  and  add,  or  triple 
~~  each  and  add,  until  the  sum  is  greater 
than  the  distance  between  the  axes,  as 


/B 


FIG.  6. 


FIG.  5. 

AB,  Fig.  5.  Then  with  AB  as  a  radius  in  the  dividers  place  one 
point  at  A  and  strike  the  arc  intersecting  at  B.  Draw  a  straight 
line  AB,  and  lay  off  on  this  line  from  A  the  greater  part  of  the 
.sum  AC,  and  from  B  the  lesser  part,  giving  the  point  C.  Through 
C  draw  the  line  of  contact.  Then  the  perpendicular  distances 
from  the  axes  A  and  B  upon  the  line  of  contact  will  be  the  radii  of 
the  wheels,  which  will  be  in  simple  proportion  with  the  distances 
BC  and  A  C. 

Second.  Contact  Outside  the  Axes. — Here  also  we  take  A 
and  B  for  the  axes,  and  C  the  line  of  contact,  the  three  lines  being 
parallel.  ^ 

With  the  same  example  of  velocity-ratio 
as  before,  take  the  difference  of  the  values 
5  and  2,  or  3  ;  or  twice  or  more  times  these 
values  if  3  is  too  small.  Then  with  the 
dividers  opened  to  the  difference  found, 
place  one  point  of  the  dividers  at  A  and  strike  an  arc  intersecting 

at  B,  and  through  AB  draw  the 
straight  line  A  C,  extended  as  far  as 
necessary.  Then  with  B  as  a  center 
and  the  lesser  of  the  above  values  lay 
off  BC;  and  the  greater  should  equal 
AC.  Now  drawing  the  line  of  con- 
tact through  C,  parallel  to  A  and  B, 
we  have  the  radii  as  perpendiculars 
from  A  upon  C,  and  from  B  upon  C. 
II.  Axes  Meeting.  Take  A  0  and 
BO  as  the  axes,  meeting  at  0.  The 
diameter  of  the  wheel  A  may  evident- 
ly be  any  line  CAE,  perpendicular 
to  its  axis  AO,  and  the  radius  A  C  = 
R.  Then  the  diameter  of  the  mat- 
ing wheel  must  be  CF,  and  the  radius  BC  =  r. 

To  determine  the  relation  between  the  radii  of  the  wheels  and 


F 


FIG.  7. 


CIBCULAK   ROLLING   WHEELS. 


their  angular  velocities,  lay  off  the  velocity  of  A  on  the  axis  of  A, 
and  the  velocity  of  B  on  the  axis  of  B,  as  On  and  Ob  respectively, 
and  complete  the  parallelogram  Oacb  by  drawing  ac  and  be.  Also 
draw  cd  and  cc  perpendicular  to  the  axes,  and  we  have  the  triangle 
ucd  similar  to  bee,  the  angle  cad  being  equal  to  cbe.  Then 

F :  v  : :  Oa  :  Ob  : :  be  :  ac  : :  ce  :  dc  : :  r  :  R. 
Whence  the  velocity-ratio 

Y  -L. 

•v       R 

proving  the  correctness  of  the  parallelogram  for  conveniently  locat- 
ing the  line  of  contact  when  the  velocity-ratio  and  angle  between 
the  axes  are  given. 

If  another  pair  of  circles  tangent  to  each  other  be  drawn  on  the 
axes  A  and  B,  as,  for  instance,  those  tangent  at  the  point  C'  on 
the  line  OC,  the  ratio  of  the  radii  of  this  pair  will  be  the  same  as 
that  of  the  circles  tangent  at  C,  as  is  plain  from  the  geometry  of 
the  figure. 

Hence  correctly  rolling  surfaces  for  this  case  must  be  cones 
with  their  vertices  in  common  as  at  0. 

In  the  above  the  contact  is  between  the  axes,  which  are  situated 
to  intersect  at  an  angle  of  less  than  90  degrees. 

When  the  angle  between  the  axes  is  greater  than  90  degrees 
there  seems  to  be  a  second  case,  as  if  the  contact  were  outside  this 
angle  as  shown  in  Fig.  8.  But 
the  principles  are  all  the  same, 
and  the  wheels  are  readily  made 
in  practice. 

The  wheel  A  of  Fig.  7  is  shown 
at  A',  Fig.  8,  so  that  the  cone  B 
may  have  a  mate  AC  or  A'C'  on 
the  axis  A. 

A  peculiar  result  is  obtained 
when  the  line  OC  is  perpendicular 
to  AO,  the  wheel  A  being  a  plane 
circular  disk  instead  of  a  cone. 
But  0,  the  intersection  of  the 
axes,  is  the  common  vertex  of  all 


_  C 


PIG.  8. 


the  rolling  surfaces  AA'B,  etc.,  in  every  instance  of  correct  rolling 
contact. 


10 


PEINCIPLES   OF   MECHANISM. 


Some  old  mill  gearing  made  wholly  of  wood  ignores  the  above- 
conic  forms  of  the  theoretic  surfaces,  where  one  wheel  has  teeth 
upon  its  side  and  the  other  upon  its  edge. 

III.  Axes  Crossing  Without  Meeting. — For  convenience,  take  the 
vertical  projections  of  the  axes  and  the  line  of  contact  of  the  roll- 
ing surfaces  as  here  shown,  parallel  to  the  "  ground  line  "  of  the  figure 
This  line  of  contact  must  be  a  straight  line,  and  tin  element  of  each 
of  the  surfaces. 

If  the  surfaces  have  a  possible  existence  for  this  peculiar  case,, 
it  is  plain  that  the  form  of  each  may  be  conceived  to  be  described 

FIG.  9. 


FIG.  10. 

by  imagining  the  line  of  contact  to  be  fixed  to  the  one  axis  and  to- 
be  revolved  about  it,  thus  generating  that  surface,  first  for  the  one, 
and  then  for  the  other.  The  form  of  surface  thus  described  is  the 
liyperloloid  of  revolution  with  axes  AA'  and  BB',  Fig.  9.  The 
smallest  diameters  will  be  at  the  common  perpendicular,  or  short- 
est distance  between  the  axes  A1 '  B* ',  which  perpendicular  is  inter- 
sected at  0'  by  the  line  of  contact  C'0'6. 

To  locate  the  line  of  contact  in  the  horizontal  projection,  imag- 
ine the  surfaces  to  be  extended  to  infinity,  where  they  become  roll- 
ing cones.  Thus  we  can  locate  the  line  of  contact  in  Fig.  10  to 


CIRCULAR    ROLLING    WHEELS.  11 

correspond  with  the  velocities  F  and  v  as  in  conic  wheels,  Fig.  7,. 
laying  off  Fon  the  axis  of  A,  and  v  on  the  axis  of  B,  as  shown,  and 
completing  the  parallelogram  with  CO,  the  diagonal  on  the  line  of 
contact  in  Fig.  10  when  sufficiently  extended.  In  the  drawing,  it 
is  most  convenient  to  place  the  line  of  contact  CO JV  parallel  to  the 
ground  line,  as  in  Fig.  10. 

To  locate  the  line  of  contact  in  Fig.  9:  -First,  it  is  plain  that  it 
will  be  parallel  to  the  axes  in  order  to  meet  the  case  of  rolling 
cones  of  infinite  extent.  Second,  to  find  the  point  0'  which  di- 
vides the  common  perpendicular  A' B'  into  segments,  or  gorge-cir- 
cle radii/draw  A'C'  and  B'G'  parallel  to  the  respective  axes  A  and  B 
of  Fig.  10.  Through  the  intersection  C',  Fig.  9,  and  parallel  to  the 
axes  draw  the  line  of  contact.  Thus  all  the  essential  lines  of  Figs. 
9  and  10  become  located,  and  the  hyperboloids  can  be  drawn  in  as 
shown,  using  the  axes  A  and  B,  Fig.  10,  as  geometric  axes,  and  the 
line  of  contact  OCN  as  an  asymptote  for  the  hyperbolic  curves. 
Afterwards  Fig.  9  can  be  completed  by  the  theory  of  projections. 

The  correctness  of  this  construction  for  the  triangle  A' B' C'  is- 
shown  by  aid  of  the  end  view  at  the  left  of  Fig.  10,  obtained  by 
passing  a  plane,  FG,  perpendicular  to  the  line  of  contact,  and  re- 
volving it  to  1JMN,  in  which  some  point  K  is  where  the  line  of 
contact  pierces  this  plane.  In  this  cross-section,  the  point  K  is  the 
point  of  contact  of  the  rolling  hyperboloidal  surfaces  as  intersected 
by  the  plane,  in  curves  tangent  at  K,  as  shown.  As  /and  /are- 
centers  of  rolling  motion  in  this  plane,  with  IK  and  JTTas  radii, 
K  must  be  on  a  straight  line  from  /  to  J9  which  locates  the  line  of 
contact  as  between  ^Yand  M,  or  between  A'  and  B'  as  well;  since 
the  line  of  contact  KC,  O'C'  is  parallel  to  the  projection  of  the 
axes  in  Fig.  9.  But  the  triangle  A'B'C'  is  similar  to  FGO,  and 
INK  to  JMK,  so  that  A'O'  +  B'0f  =  NK  +  MK,  proving  true 
the  construction  Fig.  9  for  the  position  of  C'O'. 

It  may  be  noted  that  a  system  of  hyperboloids  adapted  for  roll- 
ing upon  each  other  in  pairs  interchangeably,  may  be  drawn  by 
making  K,  CO  the  line  of  contact,  0  the  point  of  intersection  of  all 
the  axes  in  plan,  U  the  locus  of  all  their  intersections  with  the 
plane  FG,  all  axes  being  parallel  in  their  projections  Fig.  9. 

A  notable  point  in  the  history  of  this  case  is  that  the  early 
writers,  from  analogy  with  the  case  of  bevel  wheels,  assumed  that 
the  radii  Rf  and  r'  were  in  the  simple  inverse  ratio  of  the  angular 
velocities.  The  French  author  Belanger,  however,  gave  an  ex- 
tended and  rigorous  proof  to  the  contrary,  agreeing  with  the  above 


12  PRINCIPLES   OF   MECHANISM. 

graphical  construction  A'B'C'  of  Professor  Rankiue.  To  show 
this  construction  to  be  correct,  suppose  the  wheels  to  make  a  small 
turn  in  rolling  contact  so  that  the  line  of  contact,  00  produced, 
makes  a  small  displacement  from  OO  to  ab,  Fig.  10.  Then  Oa 
may  be  regarded  as  the  horizontal  projection  of  the  corresponding 
movement  of  the  point  0'  in  the  circle  A'O'.  Likewise  Ob  may  be 
the  plan  of  the  corresponding  movement  of  0'  in  the  circle  B'O'. 
But  a  and  b  should  be  in  a  line  parallel  to  CO,  as  required  for 
no  lateral  slipping  of  the  surfaces;  longitudinal  slipping  being  nec- 
essarily allowed  to  provide  for  the  endwise  sliding  of  the  teeth  of 
these  wheels  while  engaged,  and  in  action,  the  teeth  being  laid  out 
to  coincide,  in  position,  with  a  series  of  lines  of  contact,  or  straight 
elements  of  surface  properly  distributed  around  the  hyperboloids 
as  explained  under  Teeth  of  Wheels. 

Thus  conditioned,  we  see  that  the  triangle  Oab  is  similar  to 
A'B'C'. 

To  prove  this  construction  correct  we  have  from  Fig.  10 

R'  V  =  Oa,     r'v  =  Ob,     Oa  cos  aOd  —  Od  =  Ob  cos  bOd, 
and  from  the  parallelogram  0  C  we  have 

V  sin  eOC  —  V  sin  aOd  =  v  sin  f  00  —  v  sin  bOd, 
whence 

R'  __  v .  Oa        v  cos  bOd  _  sin  aOd       cos  bOd  _  tan  aOd 
~7  "  ~T7Ob  =~'  V  cos  aOd  ~  Sn.~bO'd'  X  cos  aOd  ~  tan  bOtf 

a  relation  that  must  exist  between  R',r',  v,and  V,  as  consistent  with 
the  allowed  longitudinal  sliding  along  ab,  but  not  lateral  slipping 
of  surfaces. 

From  Fig.  9,  as  constructed,  we  obtain 

O'C'  tan  A'C'O'  =  R',     O'C'  tan  B'C'O'  =  r', 
whence 

^  _  tan-4^0'      tan  aOd 
r'  ~~~  ianTB'C'6'  ~~  tan  bOd' 

ihe  same  relation  as  obtained  from  Fig.  10  as  necessarily  con- 
structed. Hence  the  construction  for  A'O'B'  as  described  is  the 
correct  one. 


CIRCULAR  ROLLING   WHEELS.  13 

These  wheels,  sometimes  called  rolling  hyperboloids,  can  never 
serve  as  practical  rolling  surfaces  or  friction  wheels,  from  the  fact 
that  the  unavoidable  longitudinal  sliding  induces  lateral  slip;  but 
in  skew-bevel  gearing  they  are  entirely  practical.  To  determine 
the  longitudinal  slip,  compare  ab  with  Oa  or  Ob,  these  being  cor- 
responding values  of  this  slip,  the  movement  of  the  end  of  R',  or 
of  r',  respectively, 

INTERMITTENT   MOTIONS. 

These  might  have  been  classified  with  respect  to  the  relation  of 
axes,  as  parallel;  meeting;  or  not  parallel  and  not  meeting;  but  it  is. 
believed  advisable  to  classify  them  with  regard  to  their  peculiarities,, 
and  treat  them  here  together. 

Tliis  name  is  given  to  movements  in  rolling  contact  where  the 
driver  A,  Fig.  11,  moves  continuously  while  the  follower,  or  driven 
piece  B,  has  periods  of  rest  and  motion. 

The  class  here  considered,  of  velocity-ratio  constant,  includes 
such  of  these  movements  as  have  circular  rolling  arcs,  here  dotted 
as  pitch  lines  for  the  teeth. 

To  be  thoroughly  practical,  the  driven  piece  should  be  positively 
locked  when  idle,  and  be  stopped  and  started  in  gradation  of 
motion  to  avoid  shocks  and  breakage. 

Spring  checks,  friction  clamps,  etc.,  should  never  be  relied 
upon,  as  a  momentary  change  in  speed  may  prevent  the  spring 
from  properly  catching  in  its  notch,  or  the  friction  clamp  from 
wear  may  get  out  of  adjustment  and  fail  to  act.  Serious  breakages 
have  been  known  to  occur  in  machines  for  making  wire  hooks  for 
hooks  and  eyes,  where  this  movement  has  been  used  with  a  friction 
clamp  instead  of  positive  locks. 

These  movements  possessing  positive  locks  are  made  in  at  least 
two  ways  which  are  thoroughly  practicable. 

First,  with  the  locking  arc  of  greater  radius  than  the  rolling 
arc,  and  provided  with  easement  spurs. 

Second,  with  the  locking  arc  of  smaller  radius  than  the  rolling 
arc,  and  provided  with  easement  segments. 

First.  Here  the  locking  arc  JK  on  the  wheel  A,  Figs.  11  and  12, 
is  given  a  larger  radius  than  the  rolling  arc  DHE,  in  order  that 
the  curves,  EJ  and  DK,  of  varying  radius  may  act  upon  the  saddle 
curve  FG  while  the  first  point  F  or  the  rolling  arc  of  B  is  moving 
into  engagement  with  the  first  point  of  the  rolling  arc  EH,  to- 
prevent  the  wheel  B  from  moving  out  of  place  in  the  one  direction^ 


14 


PKLtfCIPLES  OF   MECHAKISM. 


the  spurs  L  and  ^preventing  derangement  in  the  other  direction. 
The  spurs  L  and  N  are  on  the  front  side  of  the  wheels,  while  the 
spurs  P  and  R  are  on  the  opposite  sides,  and  dotted. 


H 


FIG.  11. 


In  Fig.  11  the  pin  N  moves  toward  the  spur  L,  fastened  upon 
JBj  striking  it  first  as  the  locking  arc  JGK  begins  to  fall  away,  the 
latter  being  cut  away  more  and  more  toward  E  so  that  B  is  just 
allowed  to  turn  with  but  slight  backlash  until  the  pin  acting  against 
the  spur  has  thus  brought  the  initial  point  of  the  rolling  arc  FIto 
contact  with  the  initial  point  of  the  rolling  arc  EH,  when  rolling 
action  begins  and  the  spurs  L  and  N  go  out  of  use  for  that  revolu- 
tion. When  the  rolling  has  continued  through  H  to  D,  the  lock- 
ing arcs  and  spurs  again  come  into  action  until  B  is  locked  again 
in  the  position  shown,  it  having  made  one  revolution  while  A 


CIRCULAR  ROLLING  WHEELS. 


15 


made  but  part  of  one,  B  remaining  idle  and  locked  while  A  com- 
pletes its  revolution. 

It  will  be  seen  that  the  pin  strikes  the  spur  quite  abruptly  in 
Fig.  11  unless  the  spur  is  considerably  inclined,  ani  starts  B  snd- 


FIG.  12. 

denly,  so  that  in  high  speed  the  pin  or  spur  may  be  broken,  while 
in  Pig.  12  the  two  spurs,  LM  and  NO,  extending  nearly  to  the 
axis  of  motion  of  the  driver  A,  start  B  slowly  at  first  and  with 
greatly  reduced  shock,  but  as  the  contact  between  the  spurs  moves 
outward,  the  wheel  B  is  accelerated  until  F  meets  E  when  rolling 
contact  begins.  By  placing  a  pair  of  spurs  on  each  side  of  the 
wheels,  they  are  adapted  for  running  either  way ;  or  if  in  one  direc- 
tion only,  it  is  important  to  have  one  pair  for  accelerating  B  at 
starting,  and  another  for  retarding  it  gradually  while  approaching 


16 


PRINCIPLES   OF    MECHANISM. 


the  locking  arc.  Thus  the  length  of  the  spurs  may  be  fixed  in  a 
particular  case  by  the  judgment  of  the  designer. 

The  rolling-circle  arcs  of  these  wheels  are  what  bring  them 
under  the  present  topic  of  rolling  contact  and  constant  velocity- 
ratio.  The  spurs  will  usually  work  by  sliding  contact,  though 
they  may  act  by  rolling,  but  with  velocity-ratio  varying,  and  their 
forms  may  be  studied  under  their  proper  headings.  (See  Fig.  141.) 

Second:  with  locking  arc  smaller  in  radius  than  the  rolling 
arc,  as  in  Fig.  13,  where  B  is  shown  as  approaching  the  locked 


FIG.  13. 


position,  and  with  £Tand  /the  rolling  arcs.  As  E  approaches  Fit 
glances  and  slides  on  F  in  such  a  way  as  to  start  B  slowly,  and 
with  acceleration  until  the  initial  points  of  the  rolling  arcs  H  and 
/  come  to  contact  when  rolling  begins  with  velocity-ratio  constant 


CIRCULAR  ROLLING   WHEELS.  17 

through  the  rolling  circular  arcs.  When  D  approaches  G,  the 
reverse  action  occurs  until  B  is  again  locked  on  the  locking  arc  as 
shown  to  soon  occur. 

In  the  details  of  these  wheels,  the  locking  arcs  are  cut  away  to 
some  extent,  as  shown  near  E  and  Z>,  to  allow  the  points  F  and  G 
to  turn  while  passing  from  lock  into  action,  and  B  is  cut  away 
somewhat  in  the  lock  between  F  and  G.  Also  the  circular  rolling 
arcs  /fand  /have  easements  at  their  ends,  thus  introducing  short 
non-circular  rolling  arcs  at  ^and  D,  to  be  set  with  gear  teeth,  when 
they  are  laid  out  for  the  rolling  arcs  treated  as  "  pitch  lines "  for 
teeth. 

Fig.  13  has  the  advantage  of  simplicity,  A  and  B  being  each  in 
practice  cast  in  one  piece,  while  Figs.  11  and  12  require  spurs, 
pins,  etc.,  which  usually  must  be  made  and  put  on  with  screws  or 


FIG,  15. 

FIG.  14. 

rivets.  The  form  Fig.  13  is  used  on  certain  binders  of  reaping 
machines. 

In  these  figures  there  is  but  one  locking  arc,  or  lock,  to  each 
wheel,  so  that  A  makes  as  many  revolutions  in  a  given  time  as  B. 
But  it  is  readily  seen  that  A  may  have  several  rolling  and  locking 
arcs  in  alternation  in  a  circumference;  likewise  B  may  have  several 
rolling  arcs  and  locks  in  its  circumference,  and  the  number  differ- 
ing from  that  of  A,  as,  for  instance,  A  may  have  three  and  B  two.. 

These  wheels  for  intermittent  motion  may  be  made  with  axes 
meeting,  or  as  not  parallel  nor  meeting,  under  the  principles  for 
those  cases  as  already  laid  down  except  for  the  easement  spurs,  and 


18  PRINCIPLES   OF   MECHANISM. 

segments,  which,  whether  acting  by  rolling  or  sliding  contact,  may 
be  treated  by  principles  considered  later,  which  see. 

In  practice,  the  rolling  arcs  in  the  above  intermittent  wheels 
must  be  set  with  gear  teeth  as  above  mentioned,  the  proper  con- 
struction of  which  teeth  will  be  considered  later  under  Teeth  of 
Wheels. 

To  give  a  clearer  idea  of  these  wheels  when  complete  with 
teeth,  locks,  spurs,  easements,  etc.,  photo-process  copies  are  here 

given,  where  Fig.  14  is  to  represent 
Fig.  12  above,  as  complete  and  ready 
for  action.  The  rolling  arcs  of  Fig.  12 
are  here  provided  with  teeth. 

In  Fig.  15  is  illustrated  a  pair  of 
wheels  explained  in  Fig.  13,  where  the 
circular  rolling  arcs  are  terminated 
with  short  portions  of  non-circular 
arcs  or  easements,  the  teeth  being 
extended  over  both.  The  distance 
between  centers  is  about  4-^  inches. 

In  Fig.  16  we  have  the  same  as  in 
FlG-  16>  Fig.  15  turned  into  bevel  wheels  with 

axes  meeting  at  an  angle  of   30  degrees,  though  the  angle  may 
be  90  degrees  more  or  less. 

CIRCULAR   ALTERNATE   MOTIONS. 

The  few  examples  for  this  case  are  deferred  to  page  82. 


CHAPTER  II. 
ROLLING  CONTACT  OF  NON-CIRCULAR  WHEELS. 

TELOCITY      RATIO      VARYING. — PITCH     LINES     OF      NON-CIRCULAR 

GEARS. 

THESE  may  be  complete,  that  is,  filling  the  full  circle  of  360  de~ 
grees,  so  as  to  admit  of  continued  rotation,  one  revolution  after  an- 
other, or  they  may  be  incomplete  or  sectoral. 

This  subject  may  very  properly  be  regarded  as  the  general 
theory  of  pitch  lines  of  gearing;  and  as  it  is  a  subject  of  much  im- 
portance it  will  be  treated  at  some  length. 

The  axes  of  motion  are  supposed  to  be  fixed  in  position,  and^ 
as  in  circular  wheels,  there  are  three  cases,  viz.:  axes  parallel;  axes 
meeting;  and  axes  not  parallel  and  not  meeting. 

CASE  I.    AXES  PARALLEL. 

Here,  as  in  circular  wheels,  the  contact  point  may  be  between 
or  outside  the  axes. 

The  following  notation  will  be  convenient  for  use: 
A  =  the  driving  wheel,  or  axis. 
B  =  the  driven  wheel,  or  axis. 
C  =  the  point  of  contact  of  wheels. 
c  =  the  distance  between  axes. 
R  =  the  radius  of  wheel  A,  and  =  AC. 
r  =  the  radius  of  wheel  B,  and  =  BG. 
V=  angular  velocity  of  wheel  A. 
v  =  angular  velocity  of  wheel  B. 
S  =  stated,  or  definite,  portion  of  arc  of  A. 
s  =  stated,  or  definite,  portion  of  arc  of  B. 
The  essential  conditions  for  true  rolling  contact  of  these  wheels 
are: 

First.  The  axes  must  be  fixed  at  a  distance  c  apart. 
Second.  The  point   of  contact  must  remain  on  the  line    of 
centers. 

19 


20  PRINCIPLES   OF   MECHANISM. 

Third.  The  sum  of  any  pair  of  radii  must  equal  a  constant, 
»    or  R  +  r  =  o  =  const. 

Fourth.  Arcs  having  rolled  truly  in  mutual  contact  are  equal, 

or  8  =  s. 

For  convenience  in  the  study  of  these  wheels,  both  A  and  B 
may  be  regarded  as  consisting  of  elementary  or  small  sectors  and 
be  treated  in  mating  pairs,  with  limiting  radii  in  pairs,  and  arcs  in 
pairs,  etc.  Thus  a  pair  of  radii  will  always  be  R  +  r  —  c,  and  a 
pair  of  arcs  will  be  8  =  s,  the  capitals  belonging  to  driver  A,  and 
the  small  letters  to  driven  wheel  B. 

Special  Cases  of  Non-  Circular  Wheels. — There  are  but  five 
simple  and  readily  demonstrated  cases  of  correct-working  non- 
circular  wheels,  viz. : 

First.  A  Pair  of  Logarithmic  Spirals  of  the  Same  Obliquity- 
(Sometimes  called  Equiangular  Spirals.) 

Second.   A  Pair  of  Equal  and  Similar  Ellipses. 

Third.  A  Pair  of  Equal  and  Similar  Parabolas. 

Fourth.  A  Pair  of  Equal  and  Similar  Hyperbolas. 

Fifth.  Transformed  Wheels. 

First.     Equal  Logarithmic  Spirals. 

The  leading  characteristic  of  the  logarithmic  spiral  is  that  it 

has  a  constant  obliquity  through- 
out. 

Taking  A,  Fig.  17,  as  the  ori- 
gin, or  pole,  of  the  spiral  A,  the 
angle   A  CD,   between   the   radius 
jj  vector   AC   and    normal    CD,  is- 
called  the  obliquity  of  the  spiral, 
its   value  being  the  same  for  all 
points  of  this  spiral.     If  a  second 
17-  spiral,  B,  of  the  same  obliquity  be 

placed  in  contact  with  the  first,  as  shown  in  Fig.  17,  the  angle 
BCE  being  equal  to  the  angle  ACD,  the  contact  point  C  will  be 
found  to  be  situated  on  the  straight  line  AB,  joining  the  poles,  for 
whatever  point  of  contact  between  the  spirals. 

Then  taking  the  points,  or  poles  A  and  B  as  fixed  axes  of 
motion,  the  spirals  will  roll  upon  each  other  without  slipping,  G 


ROLLING   CONTACT   OF   NON-CIRCULAR   WHEELS. 


21 


being  the  point  of  contact  and  AB  the  line  of  centers,  and  the  con- 
ditions for  rolling  contact  of  non-circular  wheels  are  satisfied. 


TO   CONSTRUCT   THE    LOGARITHMIC    SPIRAL. 

In  the  logarithmic  spiral,  for  equal  angles  about  the  pole,  the 
radii  vectors  are  in  geometric  progression.     Draw  two  straight  lines 


FIG.  18. 

from  0,  Fig.  18,  and  a  line  ak  and  kb.    Then  draw  bl  parallel  to 

ak,  Ic  parallel  to  kb,  cm  parallel  to  Ib,  etc.,  as  far  as  desired.     Then 

it  is  found  that  the  lines  Oa,  Ob,  Oc,  Od,  etc.,  are  in  geometrical 

progression,  or  form  a  geometric  series.     The  same  is  true  of  Oj, 

Ok,  01,  etc. ;  or  of  aj,  bk,  cl,  etc.     Hence  any  one  of  these  systems 

of  lines  may  be  used  for  radii  vectores  of  the  spiral.     Through  0, 

Fig.  19,  draw  a  series   of  lines  at 

equal   intervening  angles,  and  lay 

off  any  one  of  the  above  series  of 

lines  as  Oa,  Ob,  Oc,  etc.,  and  trace 

the  curve    abc,   etc.,  through   the 

points     thus    determined.       This 

curve  is  a  logarithmic  spiral,  the 

construction  making  it  evident  that 

the  obliquity  is  constant. 

It  is  difficult  to  construct  a  spiral,  as   above,  that  will  pass 
through  two  given  located  points  as  a  and  c  in  Fig  20.     For  this 
case  draw  a  semicircle  to  a  diameter  Aa  -f-  Ac 
as  in  Fig.  21,  and  the  perpendicular  Ab  will  be 
the  radius  which  bisects  the  angle  a  Ac,  Fig.  20, 
thus  determining  the  point  b  in 
the  curve  abc.     Then  a  similar 
construction  of  Fig.  21  for  the 
A     "a  sector  aAb  will  give  the  radius 
FIG.  20.  Fl°-  21-          Ad ;  and   another  construction 


FIG.  19. 


PELNCIPLES   OF   MECHANISM. 


the  radius  ae,  etc.,  in  continual  bisection  of  the  angles  until  the 
desired  number  of  points  in  the   spiral  ac 
E  is  obtained. 

LENGTH   OF   THE   LOG-SPIEAL   AEC. 

I  Taking  0  as  the  pole,  OD  and  OE  as  the 
limiting  radii  vectores  of  the  log-spiral  sec- 
tor ODE,  draw  a  tangent  EF  at  E.  Make 

F  OG  =  OZ>,  and  draw  GF  perpendicular  to 
OE.  Then  EF=  the  arc  />^  of  the  spiral. 
For  a  small  portion  dEg  this  appears  evi- 
dent, as  dE  and  fE  are  equal;  and  the  same 
at  the  limit  of  every  short  arc.  From  this  it 
appears  that  EH=  the  arc  of  the  spiral 

H  from  the  pole  0,  to  E,  OH  being  perpen- 
FIG.  22.  dicular  to  OE. 

APPLICATIONS. 
SECTORAL   LOG-SPIEAL  WHEELS. 

Fig.  23  shows  a  mating  pair  of  log-spiral  sectoral  wheels,  where 
the  arcs  and  also  the  differences  of 
the  limiting  radii  of  the  sectors  are 
equal;  AB  being  equal  the  sum  of 
a  pair  of  radii. 

The  velocity-ratio  for  the  posi- 
tion of   the  wheels  shown  in  the 

.      V     BC      r 
figure  is    -~=-m=15, 


FIG.  23. 


D 


R  and  r  being  the  pair  of  radii  to 
the  point  O. 

LOG-SPIEAL  LEVEES. 

Fig.  24  illustrates  the  application 
of  log-spiral  levers,  where  a  rod  D 
is  thrust  forward  or   back  by  mov- 
ing handle  E,  the  levers  working  by 
FIG.  24.  rolling  at  C. 

Another  application  is  shown  in   Fig.   25  to   a  wire-cutter, 
which  in  practice  has  been  found  to  work  well. 

A  possible   application    to  weighing-scales    is    illustrated    in 
Fig.  26. 


ROLLIHG   CONTACT   OF   ^OK-CIRCULAR   WHEELS.  23 

In  Fig.  27  is  shown  a  pair  of  "  wipers"  in  steam-engine  valve- 


FIG.  25. 


FIG.  26. 

gears  of  the  log-spiral  order.     Here  the  center  B  is  at  an  infinite 
distance  away  because  the  driven  spiral  moves 
in  a  straight  line,  being  supported  on  a  rod 
sliding  in  guide-bearings  DE.     The  spiral  B 
in  this  case  becomes  a  straight  line  as  shown. 

The  velocity-ratio  is  that  of  the  linear 
velocity  of  sliding  of  B,  and  of  the  angular 
velocity  of  A,  or  V ;  and  we  have  linear 
velocity  of  B  =  AC.  V,  or  the 

linear  velocity  of  B 
velocity-ratio  =  -  — ^ —  =  A  U.  FIG.  27. 

We  note  that  the  velocity  of  B  is  the  same  as  that  of  the  point  O 
as  revolving  about  A  ;  since  linear  velocity  of  B  =  AC  X  V. 

COMPLETE   LOG-SPIRAL   WHEELS. 

First.  Symmetrical  Uniloled  Wheels. — In  this  case  each  wheel  is 
composed  of  two  equal  sectors  of  the  spiral,  each  sector  being  one 
of  180  degrees,  as  shown.  These  admit  of  continuous  rotation  of 
A  and  B  with  a  variable  velocity-ratio,  for  which,  at  any  instant, 

V  _BC 
v  ~  AC' 

If  the  velocity  of  A  is  constant,  that  of  B  will  be  variable. 


PRINCIPLES   OF   MECHANISM. 


Wheels  like  these  are  termed  lobed  wheels,  because  extended  out 
to  one  side  in  a  lobe.     The  wheels  in  Fig.  28  are  unilobed. 


FIG.  28. 

Second.  Unsymmetrical    Unilobed   W7ieels.—Ke*v  each  wheel 
consists  of  a  pair  of  unequal  sectors  as  in  Fig.  29. 


FIG.  29. 

1.  By  aid  of  two  log-spirals.  In  this  case  construct  one  log- 
spiral  as  in  Fig.  19,  say  right-handed,  and  a  second  one,  left- 
handed,  on  a  transparent  paper.  By  placing  one  over  the  other, 

with  poles  coincident,  one  may  be 
turned  around  on  the  other  until  the 
desired  size  is  seen  lying  between  two 
consecutive  intersections,  as  in  Fig. 
30.  Thus  ADE  may  be  chosen,  or 
AEF.  If  the  first  is  too  small  and 
the  second  too  large  the  transparent 
tracing  may  be  turned  to  another 
position  over  the  other,  changing  the 
wheels  to  the  desired  size. 

X2.  By  assumed  angles  and  radii. 
Here  proceed  as  by  the   process   of 
FIG.  30.  Figs.  20  and  21,  where  the  radius  Ab 

divides  the  angle  a  Ac  and  is  made  a  mean  proportional  between  Aa 


LOGARITHMIC   SPIRAL   WHEELS.  25 

and  Ac,  as  in  Fig.  21.  Again,  find  a  mean  proportional  between 
Aa  and  Ab  for  a  radius  equally  dividing  the  angle  aAb,  etc.,  etc., 
for  as  many  points  in  ab  as  desired.  Likewise  for  the  arc  be. 

Third.  Multilobed  Wheels. — 1.  By  reduction  of  angles.  Sup- 
pose a  series  of  equidistant  radii  be  drawn  in  Fig.  28  or  29,  and  that 
the  angles  thus  found  be  reduced  by  one  half.  Fig.  29  would  then 
be  changed  to  Fig.  31,  giving  the  half  of  a  2-lobed  wheel  ;  the 


other  half  being  like  it.  This  wheel  would  work  correctly  with 
another  like  it,  making  a  pair  of  2-lobed  wheels. 

Again,  by  reducing  the  angles  of  Fig.  29  to  a  third  their  value, 
the  third  of  a  3-lobed  wheel  would  be  obtained,  from  which  the  full 
wheel  could  be  obtained  by  uniting  three  copied  sectors.  Likewise 
a  4-lobed,  5-lobed,  etc.,  wheel  could  be  produced. 

These  wheels  would  work  together  in  pairs  of  equal  lobes,  but 
not  interchangeably. 

2.  By  assumed  angles  and  limiting  radii.  In  Fig.  31  lay  off 
AE  —  AF  as  maximum  radii,  and  AD  as  a  minimum  radius. 
Bisect  the  angle  DAFby  AG  and  find  the  length  of  AG  as  in  Fig. 
21,  thus  obtaining  a  point  G  in  the  arc  DGF.  Similarly  bisect 
GAP  by  AH,  find  AH  as  by  Fig.  21,  giving  Ha  point  in  the  curve. 
The  point  1  is  found  in  the  same  way,  and  other  points  between,  to 
the  extent  desired,  when  the  curve  PHD  can  be  drawn  in. 

Likewise  for  sector  DAE,  giving  the  half  of  a  2-lobed  wheel 
from  which  the  whole  wheel  is  readily  formed  either  by  copying 
the  second  half  from  Fig.  31  as  rights  and  lefts  with  this  half, 
making  the  wheel  symmetrical  with  respect  to  EF  as  an  axis  of 
symmetry,  or  by  swinging  the  copied  half  around  on  the  paper 
180°  and  joining  it  to  this  half  EDF,  giving  a  non-symmetrical 
wheel. 

Pairs  of  3-lobed,  4-lobed,  etc.,  wheels  may  be  thus  produced,  but 
they  would  not  work  interchangeably. 


26 


PRINCIPLES   OF   MECHANISM. 


INTERCHANGEABLE   MULTILOBED   LOG-SPIRAL   WHEELS. 

1.  As  derived  from  one  spiral. 
A  1-lobed  wheel  as  in  Fig.  28  will 
consist  of  two  equal  sectors  of  180°  each, 
as  DAGH,  Fig  32,  or  PAGE,  etc.,  ac- 
cording to  size. 

A  2-lobed  wheel  will  consist  of  four 
FIG.  32.  sectors  of  90°  each,  as  EAHG,  etc. 

A  3-lobed  wheel  will  require  six  sectors  of  60°  each,  etc. 
That  these  several  wheels  be  interchangeable  as  derived  from 
one  spiral,  Fig.  32,  it  is  necessary  that  the  sectoral  arcs  equal  each 
other,  while  the  sectoral  angles  differ. 

Thus,  in  Fig.  33,  the  arc  DCE  must  equal  the  arc  FCG,  while 


FIG.  33. 

the  angle  DAE  =  180°,  and  the  angle  BFG  =  60°. 

In  Fig.  32  the  180°  sector  DHG  gives  half  the  wheel  A,  Fig.  33^ 
while  the  sector  EAJ,  Fig.  32,  gives  a  sixth  of  the  wheel  B,  Fig. 
33,  these  sectors  in  Fig.  32  being  selected  by  trial  measurements  on 
a  drawing-board  so  that  the  arc  DHG  =  arc  EIJ.  Likewise  the 
sector  FAIy  Fig.  32,  would  make  the  eighth  of  a  4-lobed  wheel,  and 
HAE  the  fourth  of  a  2-lobed  wheel,  any  two  of  which  series  of 
wheels  from  the  1-lobed  to  the  4-lobed  and  upward  will  work  to- 
gether in  pairs  interchangeably. 

BG 

The  velocity -ratio  is  always  equal  -r-^. 

A  L> 

2.  As  derived  from  two  spirals  of  unequal  obliquity. 
In  Fig.  30  consider  the  wheel  DEG  as  a  1-lobed  wheel,  to  work 
with  which  a  2-lobed,  3-lobed,  etc.,  wheel  is  desired. 

Here  it  is  only  necessary  to  proceed  with,  say,  the  steepest  spk-al 


LOGARITHMIC   SPIRAL  WHEELS.  27 

first,  just  as  was  done  in  Fig.  32,  obtaining  sectors  with  arcs  equal 
that  of  the  corresponding  steepest  sector  of  the  one-lobed  wheel, 
and  with  angles  a  half,  a  third,  etc.,  as  great;  resulting  in  the 
steepest  sectors  of  the  2-lobed,  3-lobed,  etc.,  wheels;  then  likewise 
with  the  flatter  spiral,  resulting  in  the  flatter  sectors,  for  the 
2-lobed,  3-lobed,  etc.,  wheels;  when  the  unequal  sectors  for  a  lobe 
of  any  one  of  the  proposed  wheels  can  be  selected  and  the  wheels, 
laid  out. 

Fig.  34  is  an  example  of  a  1-lobed  wheel  mating  with  a  3-lobed 


FIG.  34. 


one  laid  out  in  this  way.    Any  two  of  the  lobed  wheels  in  the- 
series  above  described  will  roll  truly  together. 

3.  As  determined  from  proportional  sectors. 

In  Fig.  35  let  ADEH  represent  any  given  sector  of  a  lobed 


I  DA 

FIG.  35. 

wheel.  Required  a  sector  AIJ  of  a  wheel  of  three  times  as  many 
lobes  that  will  work  correctly  with  the  first-named  wheel.  In  this 
case  the  sector  AIJ  must  work  correctly  on  the  sector  ADEH. 

Now  for  three  times  as  many  lobes  in  the  new  wheel,  new  sec- 
tors of  the  new  wheel  which  mate  with  the  first  must  have  the 
sectoral  angle  one  third  that  of  the  first  wheel.  That  is,  for  the 
sector  AIJ  to  mate  with  sector  ADEH,i\iQ  angle  IAJ  must  equal 
a  third  of  the  angle  DAH,  and  it  is  plain  that  the  sector  AIJ  must 


PRINCIPLES   OF   MECHANISM. 


be  similar  to  the  sector  A DE  because  of  the  equal  obliquities.    Also 
JN  must  equal  HK,  and  IJ  equal  DEH.     Therefore 
EL  :  HK  :  :  AD  :  AI  :  :  AE  :  AJ, 

a  proportion  which  can  be  readily  constructed  graphically  as  in 
— -^    Fig.  36.     With  the  mating  sectors  ADEH  and 
AIJ  thus  found,  the  wheels  may  be  laid  out. 

If  in  the  first  wheel  the  other  sector  for  a 
H  lobe  is  not  equal  to  the  sector  ADEH,  but  tha 
two  for  a  lobe  form  an  aliquot  part  of  the  full 
circle  of  360  degrees,  then  a  similar  construction 
will  give  the  new  mating  sectors  for  the  other  parts  of  the  lobes  of 
the  new  mating  wheel. 

In  this  way  the  mating  4-lobed  wheel  is  found  for  the  given 


D 
FIG.  36. 


2-lobed  wheel  A  of  Fig.  38  as  shown  in  Fig.  37,  where  EDF  is  the 
half  of  the  given  wheel.  The  number  of  lobes  being  doubled,  the 
angles  will  be  halved,  and  the  halves  of  the  sectors  of  Fig.  37  being 
similar  to  the  new  sectors  required,  are  simply  expanded.  Thus 
the  half  AKD  is  expanded  to  the  full  new  sector  AHG,  making 
HL  =  EJ.  Similarly  the  half  of  ADF,  or  ADN  is  expanded  to 
the  full  new  sector  A  GI,  and  the  new 
4-lobed  wheel  working  with  the  former 
given  2-lobed  wheel  are  both  shown, 
mated,  in  Fig.  38. 

When  one  wheel  is  expanded  to  an 
indefinite  number  of  lobes  it  becomes  a 
straight  notched  bar,  as  in  Fig.  39,  of 
indefinite  extent,  the  sectoral  arcs  of  FIG.  38. 

which  are  straight  inclined  lines  where  A  may  be  the  driver,  O  the 
point  of  contact,  and  the  center  B  at  an  infinite  distance  away. 
The  wheel  B  thus  becomes  a  sliding  bar  DE.  The  driver  A  may 
have  unequal  sectors  when  the  notches  will  be  unsymmetrical  like 
the  teeth  of  a  rip-saw. 


LOGARITHMIC    SPIRAL   WHEELS. 

The  velocity-ratio  is  the  same  as  that  for  Fig.  27. 


FIG.  40. 


FIG.  39. 
EASEMENT   CURVES. 

One  characteristic  of  the  above  purely  log-spiral  wheels  is  the 
sharp  abrupt  intersection  points,  both  salient  and  re-entrant,  in  the 
outlines  where  the  sectors  lie 
adjacent.  These  if  desired  may 
be  eased  off  into  curves  of  any 
form  provided  they  will  roll 
properly,  as  in  Fig.  40. 

Thus  ADE  is  an  easement  F/ 
sector  mating  with  BHI,  and 
AFG  another  mating  with  BJK. 
The  easement  sectors  cannot  be 
log-spirals,  and  they  may  be  constructed  by  assuming  one,  as,  for 
instance,  DE,  and  then  the  mating  curve  ///  must  be  found.  How 
to  thus  construct  mating  curves  is  explained  in  a  future  chapter. 
To  make  such  filleted  outlines,  DE  and  FG  may  be  assumed,  and 
the  curves  D F  and  GE  drawn  in  as  log-spiral  sectors.  Then  as 
the  position  of  the  point  B  is  not  known  exactly,  an  assumption 
may  be  made  for  it  and  tested  by  constructing  a  sector  HIJK, 
which  must  be  in  proper  relation  to  the  full  circle  of  360  degrees. 

Second.     Equal  and  Similar  Ellipses. 

Fig.  41  presents  a  pair  of  equal  and  similar  ellipses  tangent  to 
each  other  at  C,  with  A,  D,  B,  and  E  focal  points.  The  ellipses 
are  intentionally  placed  in  contact  so  that  the  distance  CH  equals 
the  distance  CI,  the  points  H  and  7  being  at  the  extremes  of  the 
major  axes.  If  the  ellipses  be  now  rolled  in  mutual  contact  without 
slipping,  in  the  direction  such  that  H  and  /will  approach,  these 
points  will  meet  when,  at  the  same  time,  the  major  axes  of  the  ellipses 
will  coincide  with  one  and  the  same  straight  line  through  AB. 


30  PRINCIPLES   OF  MECHANISM. 

It  is  readily  seen  that  Fig.  41  is  symmetrical  with  reference  to 
the  common  tangent  CO,  and  that  AC  —  EG,  BC  =  DC. 

From  a  well-known  property  of  the  ellipse  AC  -f  CD  equals 
the  major  axis  of  the  ellipse,  equals  FH,  a  constant  for  all  points 
of  contact  C.  But  as  CD  always  equals  CB,  we  have  A  C  -f  CD 
=  A  C  +  CB  =  AB,  equal  a  constant,  and  equals  the  major  axis 
of  the  ellipse.  Likewise  for  BCE.  Hence  if  we  take  the  focal 


FIG.  41. 

points  A  and  B  as  axes  of  motion,  the  ellipses  will  roll  truly  in 
mutual  contact  as  non-circular  wheels. 

For  the  same  reason  that  AB  is  constant,  DE  is  also  constant, 
and  hence  pins  placed  at  D  and  E  may  be  connected  by  a  rod  or 
link  with  freedom  of  movement  about  A  and  B.  It  is  interesting 
to  note  that  the  elliptical  wheels  transmit  the  same  motion  from  A 
to  B  as  would  a  pair  of  cranks  AD  and  BE  and  a  connecting  rod 
DE. 

The  velocity-ratio  for  the  elliptic  wheels  of  Fig.  41  is  1  when 
the  extremities  of  the  minor  axes  are  in  contact  at  C,  or  when 
AC  —  BC\  and  least,  or  greatest  in  value,  when  the  major  axes 
touch  at  extremities  on  the  line  A  B. 

If  A  is  driver  and  revolves  uniformly,  then  the  fastest  move- 
ment of  B  occurs  when  H and  /are  in  contact,  and  slowest  when 
F  and  G  are  in  contact ;  hence  the  ratio 

AH 

Fastest  for  B  _  Bl_  _  AH  BG  __  AH* \ 
Slowest  f  or  B  ~  AF  ==  BI '  AF  ~~  BI* 

GB 


ELLIPTIC   WHE 


OF   THE 

UNIVERSITY 


31 


This  ratio  increases  rapidly  with  eccentricity.  For  example,  in  a 
nail- driving  device  for  a  certain  boot  and  shoe  nailing  machine 
(see  Fig.  130)  with  elliptic  pitch  lines,  AH  =  3.25  ins.  and 
BI  =  1  inch,  so  that  the  fastest  for  B  is  10.56  times  the  slowest; 
and  the  absolute  velocity  of  A  being  usually  about  300  revolutions 
per  minute,  the  slowest  rate  of  absolute  movement  for  B  is  there- 
fore r-^rr-  X  300  —  92.3  revolutions  per  minute,  while  the  fastest  is 
o»/vD 

3.25 

-^ —  X  300  =  975  revolutions  per  minute.  The  ratio  of  these  fig- 
ures is  the  same  as  by  the  formula  givena  bove,  viz.,  10.56.  In  this 
nail-driving  device  a  crank  is  attached  to  B  and  by  aid  of  a  pitman 
or  connecting  link  reciprocates  a  driving  bar,  the 
hastened  motion  of  which  bar  resembled  that  of 
a  driving  bar  accelerated  by  a  spring,  but  possessed 
the  great  advantage  of  requiring  no  buffer  to  absorb 
the  residual  terminal  shock,  the  crank  and  link 
always  preventing  the  driving  bar  from  going  be- 
yond its  predetermined  lowest  position.* 

Fig.  42  is  an  example  from  the  Putnam  Ma- 
chine Co/s  exhibit  at  the  Centennial  of  1876,  of 
elliptic  gears  used  on  a  slotting  machine  to  ap- 
proximate the  "  quick  return."  The  fastest  mo- 
tion of  the  driven  wheel  is  nearly  4.5  times  the 
slowest.  The  distance  between  centers  was  14 
inches. 

See  also  Fig.  133,  for  which  the  fastest  for  the  driven  wheel  is 
41  times  the  slowest. 


FIG.  42. 


*  A  general  expression  for  the  velocity-ratio  is  obtained  from  the  equation 
of  the  ellipse,  viz. : 

ml 


BG 


m  —  a.  cos  a* 


m  -\-  a .  cos  ff 


from  which  we  find  the 

velocity-ratio  =  %= 


m  —a.  cos  a* 


where  a  and  j3  are  the  angles  DAG  and  IBC;  and  where  2w  is  the  major  di- 
ameter and  2n.  the  minor  diameter,  a  being  %BE. 


OZ  PRINCIPLES   OF   MECHANISM. 

The  smoothness  of  outline  of  these  elliptic  wheels  for  rolling 
contact  is  favorable  for  their  use  in  many  cases  where  a  variable 
velocity  of  the  driven  wheel  is  the  main  object,  instead  of  some 
definite  law  of  motion. 

As  the  curves  here  considered  are  only  for  pitch  lines  of  gear 
wheels,  it  is  plain  that  angles  in  those  lines  are  by  preference 
avoided.  Compare  Figs  42  and  45  with  Figs.  71,  73,  and  89. 

INTERCHANGEABLE    MULTILOBED    ELLIPTIC   WHEELS. 

The  remarkable  elliptic  wheels  of  the  Rev.  H.  Holditch  are 
the  only  ones  that  may  properly  come  under  this  classification. 
These  are  called  elliptic,  not  because  made  up  of  parts  of  ellipses, 
but  because  of  the  peculiar  relation  of  this  series  of  wheel  curves 
to  a  corresponding  series  of  ellipses  as  shown  in  Fig.  43. 

Draw  the  ellipse  DPE  with  foci  A  and  M  and  center  0.     Draw 


OS  perpendicular  to  AM,  cutting  the  ellipse  at  P.  Then  for 
complete  multilobed  wheels,  lay  off  OP  in  repetition  on  OS,  giving 
points  Q,  R,  S,  etc.,  so  that  SR=RQ=QP=PO.  Then 
through  these  points  draw  confocal  ellipses  as  shown,  that  is,  with 
A  and  M  as  focal  points.  The  points  F,  H,  K,  G,  I,  etc.,  may  be 
found  by  making  OF  =  OG  =  AQ,  OH  =  01  =  AR,  etc.,  etc. 
Also  we  have  Aa  -\-  aM  =  AQ  -f-  QM,  etc.  The  full  elliptic  arc 
may  be  drawn  by  stretching  a  thread  AQMA  around  pins  at  A  and 
M,  and  a  pencil  at  Q.  Then  moving  the  pencil  either  way,  kept 
tight  against  the  thread,  the  ellipse  is  traced. 

The  ellipse  DPE  is  the  unilobe  as  at  A  in  Fig.  44.  The  bilobe 
is  obtained  from  FQG,  Fig.  43,  by  using  the  radii  AF,  Aa,  Aft,  Ac, 
etc.,  laid  off  on  lines  radiating  from  B,  Fig.  44,  at  half  the  angles 
FAa,  aAb,  bAc,  respectively,  Fig.  43,  this  giving  a  sector  CBT, 
Fig.  44,  which  is  the  fourth  part  of  the  bilobe  required. 


ELLIPTIC   WHEELS. 


33 


Similarly  the  trilobe  is  obtained  by  laying  off  the  radii  AH, 
Ad,  Ae,  etc.,  on  lines  radiating  from  B',  Fig.  44,  at  a  third  of  the 
angles  HAd,  dAe,  etc.,  Fig.  43,  thus  giving  a  sector  C'U'U,  which 


FIG.  44. 

is  the  sixth  part  of  the  trilobe;  from  which  the  wheel  can  be  con- 
structed, as  shown  in  Fig.  44. 

In  practice  it  is  most  convenient  to  make  the  angles  equal  to 
each  other  between  lines  radiating  from  A,  Fig,  43;  in  which  case 
the  angles  between  the  lines  radiating  from  B,  Fig.  44,  are  equal, 
and  a  half  as  great  as  in  Fig.  43,  and  angles  at  B',  Fig.  44,  equal  a 
third  of  those  of  Fig.  43,  etc. 

The  unilobe  A,  Figs.  43  and  44,  is  an  ellipse  as  stated.  But 
the  bilobe  is  not  an  ellipse,  though  somewhat 
resembling  it.  Again,  the  wheel  B'  is  far 
from  being  an  ellipse,  and  the  reason  these 
wheels  are  called  elliptic  is  because  of  the 
use  of  ellipses  in  Fig.  43  in  constructing 
them. 

Fig.  45  is  a  photo-process  copy  of  a  pair 
of  elliptic  wheels  like  B  and  B'  of  Fig.  44, 
where  the  rolling  curves  serve  as  pitch  lines 
for  the  teeth. 

The  demonstration  of  the  Holditch  wheels 
is  not  simple,  and  will  not  be  given  here, 
but  the  wheels  are  admitted  at  this  point 
because  of  their  direct  dependence  upon 
ellipses. 

For  the  particular  case  that  the  number  of  lobes  becomes  infi- 
nite, the  center  of  the  wheel  is  removed  to  infinity,  and  what  we 


FIG.  45. 


34 


PRINCIPLES   OF    MECHANISM. 


see  of  it  is  shown  in  Fig.  46,  where  the  curve  becomes  a  sinusoid 
LFHK,  in  which  LH '=  nn,  and  GF  =  NK  =  Vvi*  -  11*  =  AO, 
if  m  and  n  be  taken  as  semi-axes  of  the  ellipse. 


FIG.  46. 

To  construct  the  ellipse  on  a  drawing-board :  The  half  length  and 
breadth  m  and  n  having  been  determined,  and  laid  off  on  OJ  and 
01,  make  I  A  =  ID  =  m.  Then  take  any  radius  AP  and  describe 
an  arc  at  P.  Subtract  this  radius  from  2m,  and  with  this  differ- 
ence as  a  radius,  equal  DP,  describe  an  intersecting  arc  at  P. 
The  intersection  will  be  a  point  in  the  ellipse.  Any  number  of 
points  may  be  thus  found  and  the  ellipse  drawn  in. 

To  construct  the  arc  FH  —  HK,  etc.:  Take  from  a  table  of 
sines  the  values  for  every  nine  degrees  up  to  ninety  degrees,  multi- 
ply these  by  4/m2  —  ri*,  and  lay  them  off  on  ten  equidistant  perpen- 
diculars between  H  and  G  and  including  the  latter,  through  which 

points  draw  the  curve  FH,  OH  being  =  —  *. 

The  velocity-ratio  is  the  same  expression  as  that  for  Fig.  27, 
viz.:  taking  linear  velocity  of  B  as  W,  and  angular 'velocity  of  A 
as  F,  we  have  AC.  V—  W, 


*  Where  the  semi-axes  of  the  ellipse  are  m   and  n  the  equation  of  the 
curve  of  the  sinusoid  LFHK  is 

.    x  •*•  "' 

sin  —  — 
n 


found  by  integrating  the  polar  equation  of  the  ellipse  for  dx  =  rdv  and  remem- 
bering that  r  =  m  ±  y. 


PARABOLIC   WHEELS. 

W 


35 


or 


making  the  rate-  of  linear  movement  of  B  along  the  guides  M  and 
N  the  same  as  that  of  the  end  of  a  radius  AC  for  the  instant  as 
revolving  about  the  center  A. 

Third.     A  Pair  of  Equal  and  Similar  Parabolas. 

A  parabola  may  be  regarded  as  an  ellipse  so  extended  that  one 
focus  is  at  an  infinite  distance  away.  Thus  in  Fig.  41  suppose  the 
focus  B  and  D  to  be  infinitely  removed.  Then  F  and  G  are  ver- 
tices of  parabolas,  with  A  one  axis  of  motion  while  the  other,  B, 
is  at  an  infinite  distance.  In  this  case  the  motion  of  EG  is  simply 
to  slide,  as  for  B  in  Fig.  46.  These  rolling  parabolas  are  shown  in 


FIG.  47. 

Eig.  47,  where  A  is  the  fixed  axis  of  revolution  for  wheel  A,  while 
B  slides  in  guides  M  and  N. 

To  prove  that  the  parabolas  will  thus  roll  truly,  we  make  use 
of  Fig.  48  of  two  equal  parabolas  with  their  vertices  in  contact  at 
Cy  and  with  A  and  0  focal  points,  A  serving  as  an  axis  for  A,  and 
with  B  sliding  in  guides  M  and  N.  A  property  of  the  parabola 
is  that  the  radial  line  AD  makes  the  same  angle  as  does  DFwith 
the  tangent  at  D,  or  OF  with  the  tangent  at  F.  Also  a  well-known 
property  is  that  the  line  AD  equals  the  line  DO  equals  FH,  where 
OG  and  AH  are  perpendicular  to  AO.  Now  with  A9  M,  and  N 
fixed,  suppose  A  to  turn  until  D  falls  at  K,  at  the  same  time  sliding 
OB  along.  Then  Fwill  be  brought  exactly  to  K,  with  the  latter  a 


PRINCIPLES   OF   MECHANISM. 


point  of  rolling  contact  on  the  line  of  centers  A  OK,  as  shown  by 
the  dotted  lines,  because  AK —  FH  —  GD,  and  the  angles  between 
the  tangents  to  the  parabolas  at  K  and  the  line  A K  are  equal  to  the 
angle  between  AD  and  the  tangent  at  D.  The  same  is  true  for  all 


FIG.  48. 

the  points  of  contact  between  the  parabolas  of  Fig.  48,  and  hence? 
the  parabolas  of  Fig.  47  will  roll  truly  upon  each  other,  A  turning 
about  its  pin  while  B  slides  in  M  and  N. 

Fourth.    A  Pair  of  Equal  and  Similar  Hyperbolas. 
In  Fig.  49  let  LCM  and  ICJ  represent  a  pair  of  equal  and  sirni- 

M 

P/ 


FIG.  49. 


FIG.  50. 


lar  hyperbolic  sections  for  which  A  and  E  are  focal  points;  tha 
focus  A  being  taken  for  the  axis  about  which  LCM  rotates,  and  B 
the  axis  about  which  ICJ  turns,  the  latter  being  mounted  by  a  bar 


HYPERBOLIC   WHEELS.  37 

BE,  made  fast  to  it,  and  centered  by  a  pivot  at  B.  A  and  B  are 
then  centers  of  motion,  and  BA  C  the  line  of  centers,  which  must 
of  course  constantly  pass  through  the  point  of  contact  6". 

To  prove  that  these  hyperbolas  will  roll  truly,  we  refer  to 
Fig.  50  of  the  same  hyperbolas  presented  in  correct  relation  with 
their  asymptotes  OP  and  OK,  and  focal  points  A  and  E.  A  well- 
known  property  of  these  curves  is  that  the  difference  in  length  of 
any  pair  of  lines  A  C  and  EC  from  any  point  C  is  the  same  for  all 
points  C  on  the  curves,  and  is  equal  to  the  distance  FG  between 
the  vertices  and  constant.  Another  property  is  that  the  tangent 
CH  at'C  bisects  the  angle  between  the  radial  lines  AC  and  EC. 

Now  if  we  suppose  the  hyperbola  A  CF  to  be  lifted  from  the 
paper  as  drawn,  and  be  swung  around  till  HC  coincides  with 
H'G",  and  the  point  C  with  the  point  C";  then  the  hyperbolas 
will  be  in  common  tangency  at  C',  and  the  distance  from  the  new 
position  of  A  over  to  B  will  just  equal  BO'  -  AC  =  FG,  =  AB 
of  Fig.  49;  and  the  two  figures  will  otherwise  agree:  presenting 
conditions  consistent  with  true  rolling  contact. 

The  velocity-ratio  is 

V  _BC 
v  ~~~  AC' 

In  this  case  the  point  of  contact  is  outside  the  space  AB,  and 
at  any  instant  the  wheels  revolve  in  the  same  direction,  while 
in  former  cases  the  contact  is  between  the  axes  and  the  motions 
contrary. 

Fifth.    Transformed  Wheels. 

Let  Fig.  51  represent  any  pair  of  correct  rolling  non-circular 
wheels;  in  this  case  sectoral.  ^  p 

Divide  the  two  wheels  into  mat- 
ing elementary  sectors,  for  which  a  ^  /^^^\~  tui  =§|^QB 
couple  of  mating  short  arcs  may  be 
referred  to  as  a  pair  of  arcs;  a  mat- 
ing couplet  of  radii  as  a  pair  of  radii; 
and  a  mating  set  of  angles  as  a  pair 
of  angles. 

Then  the  wheels  A  and  B  may  be  transformed  in  three  ways, 
viz. : 

1st.  By  multiplying  each  of  a  pair  of  angles,  or  of  arcs,  by  a 
constant.  (Holditch.) 


38 


PRINCIPLES   OF   MECHANISM. 


FIG.  52. 


2d.  By  adding  a  constant  length  to  all  radii  of  one  wheel,  the 
arcs  remaining  constant. 

3d.  By  adding  to  one  and  subtracting  from  the  other  of  a  pair 
of  radii  a  given  quantity;  arcs  remaining  unchanged. 

In  these  changes  the  elementary,  or  small,  sectors  are  supposed 
to  have  angles  of  from  three  to  ten  degrees,  according  to  desired 
accuracy  of  result,  the  precision  of  mathematical  analysis  not  being 
expected,  though  a  sufficiently  close  approximation  to  it  is  ob- 
tained to  answer  for  most  practical  problems  arising  under  this 
proposition. 

The  second  mode  of  transformation,  applied  to  the  wheel  B  of 
Fig.  51,  by  cutting  off  BE'  from  all  radii, 
or  from  B  to  the  circle  DE,  that  is,  adding 
negatively  the  length  BB'  to  all  radii  of  B, 
and  bringing  the  cut-off  ends  of  the  re- 
maining portions  of  radii  all  to  the  point 
B',  will  give  the  pair  of  wheels  shown  in 
Fig.  52  which  will  be  found  to  answer  the 
conditions  of  true  rolling  contact. 
The  first  mode  of  transformation  applied  to  the  wheels  of  Fig. 
52,  by  multiplying  all  the  angles  by  two  for  a  simple  case,  changes 
those  wheels  to  the  ones  shown  in  Fig.  53. 
In  this  multiplying  it  is  not  necessary  to  use 
the  same  multiplier  for  all  pairs  of  angles; 
indeed  a  different  multiplier  might  be  used 
for  each  pair  of  angles. 

The  third  mode  of  transformation  ap- 
plied to  Fig.  52,  results  in  Fig.  54,  where  Aa 
is  shortened  and  Be  lengthened  the  same 
amount;  Ab  shortened  and  Bd  lengthened  the  same  amount;  Ae 
lengthened  and  Bg  shortened  the  same  amount;  and  finally,  Af 
shortened  and  Bh  lengthened  the  same 
amount,  these  changes  of  the  radii  being- 
made  without  any  alteration  of  the  lengths 
of  the  elementary  arcs  Ca,  ab,  Cc,  cd,  Ce,  Off, 
etc.  It  will  be  understood  that  the  posi- 
tions of  the  radii  will  be  changed,  that  is, 
that  the  angle  CAa  will  be  varied  as  Aa  is 
FIG.  54.  shortened,  while  the  length  Ca  is  preserved; 

and  likewise  for  the  other  angles  and  arcs. 

These  three  modes  of  transformation  are  sufficient  to  change 


FIG.  53. 


TRANSFORMED    WHEELS. 


39 


any  pair  of  correct  rolling  wheels  into  any  other  form.  Thus, 
starting  with  the  sectoral  wheels  of  Fig.  51,  suppose  it  were  sought 
to  derive  a  pair  of  complete  non-circular  wheels. 

Then,  first,  make  the  limiting  radii  of  A  equal,  the  one  to  the 
other,  also  B,  by  the  third  mode;  then  by  multiplying  pairs  of 
angles  by  a  given  quantity,  one  pair  after  another,  as  by  the  first 
mode,  we  may  make  the  total  angle  of  one  of  the  sectors  360  de- 
grees, giving  a  complete  wheel,  while  the  other  will  most  likely  not 
be  complete.  To  make  the  latter  complete  also,  apply  the  second 
mode  of  transformation  until  that  wheel  closes  with  360  degrees. 

That  the  wheels  thus  rendered  complete  shall  have  the  neces- 
sary shape  as  to  eccentricity,  law  of  varying  velocity-ratio,  etc.,  the 
wheels  may  be  examined  before  entirely  closed,  and  so  treated  by 
the  modes  of  transformation  as  to  bring  about  the  desired  law  of 
velocity-ratio  of  the  completed  wheels. 

Extended  changes  of  wheels  in  this  way  in  general  become 
tedious,  and  other  processes  on  the  drawing-board,  hereafter  ex- 
plained, will  be  preferred.  The  above  modes  of  transformation 
will,  however,  be  found  most  useful  in  making  minor  changes,  such 
as  rounding  off  a  sharp  prominence  or  depression,  as  in  Fig.  40; 
and  in  certain  special  cases. 


INTERCHANGEABLE   MULTILOBES   BY   TRANSFORMATION. 

The  transformation  of  a  pair  of  correct  working  wheels  into 
another  pair,  wider  or  narrower, 
by  the  first  mode,  is  a  compara- 
tively simple  operation. 

Thus,  in  Fig.  55,  we  have  a 
pair  of  Transformed  Elliptic 
Wheels,  mating  half-ellipses  A  and 
B,  in  dotted  lines,  in  which  the 
angles  are  reduced  by  one  half  FIG.  55. 

without  changing  the  radii  by  which  the  180-degree  sectors  are 
reduced  to  90-degree  sectors,  or  the  one-fourth  part  of  a  pair 
of  2-lobel  wheels,  as  explained  in  Willis,  page  65. 

Other  portions  of  rolling  ellipses  may  be  transformed  into 
lobed  wheels  of  intersecting  characteristics,  as  explained  in  Mac- 
Cord's  Mechanical  Movements,  page  54. 

As  a  second  example,  take  the  equal  elliptic  wheels  A  and 
B,  Fig.  56,  and  transform  A  into  a  wheel  of  two  lobes  as  at  A\ 


40 


PRINCIPLES   OF   MECHANISM. 


FIG.  56. 


by  adding   a  constant  quantity  to  all  the  radii  of   wheel  A,  the 

elementary  arcs  remaining  un- 
changed, as  per  the  second  mode 
of  transformation.  We  will  trans- 
form the  half  of  A,  the  other  half 
being  like  it  except  as  to  "  rights  and 
lefts/' 

The  amount  to  add  to  all  the 
radii  of  A  to  obtain  a  90-degree  re- 
sulting sectoral  wheel  A'C'M'  to 
give  exactly  the  fourth  part  of  the 
wheel  A'  is  not  readily  determined, 
several  trial  values  being  in  some 
cases  necessary.  In  the  present  case, 
however,  we  may  refer  to  the  Hoi- 
ditch  principle  of  Fig.  43,  from 
which  we  find  that  by  subtracting 
the  distance,  Fig.  56,  AD  from  the  distance  AE  we  obtain  EF 
for  the  amount  to  add  to  all  the  radii  CA,  GA,  HA,  etc.  To 
apply  this  conveniently,  draw  the  circle  NRP  to  center  A  with 
radius  EF. 

Then  to  construct  A'C'M*  take  A'C*  =  CN,  thus  adding  AN 
to  AC.  Also  make  A' G'  =  GQ,  and  C'G'  =  CO,  thus  determining 
the  point  G'.  For  the  point  fT  make  A'H'  =  HR,  and  G'H'  =  GH, 
thus  determining  the  point  //'.  Likewise  proceed  to  M',  making 
A'M'  =  PM.  Trace  the  curve  through  the  points  C'G'H' .  .  .  M', 
completing  the  90-degree  sector  A'C'  M',  with  which  the  complete 
2-lobed  wheel  A'  may  be  laid  off.  This  wheel  A'  will  work  cor- 
rectly with  the  wheel  B  as  a  pair  of  2-lobed  and  1-lobed  wheels,  as 
A  and  B  of  Fig  44. 

The  quarter  wheel  A'  C'M'  is  identical  with  the  Holditch  wheel 
found  by  the  principles  of  Figs.  43  and  44,  though  the  points 
G'9  Hf,  etc.,  will  not  coincide  with  the  Holditch  points  even  for  the 
same  number  of  points,  and  the  same  angle  CAG,  GAH,  etc. 

To  make  a  3-lobed  elliptic  wheel  by  this  mode  of  transformation 
make  E8  =  ED,  and  subtract  AD  from  AS,  which  difference  use 
as  the  radius  AT  of  a  circle  about  A,  from  which  measure  the  radii 
for  the  3-lobed  sector  similarly  as  in  the  2-lobed  wheel.  Combine 
these  radii  with  the  same  arcs  CG,  GH,  etc.,  for  points  in  the 
60-degree  sector  for  the  sixth  part  of  the  3-lobed  wheel. 

Fig.  28  could  be  used  as  basis  of  a  series  of  multilobed  wheels 


TRANSFORMED    WHEELS. 


41 


which  would  work  interchangeably,  spiral  arcs  and  differences  of 
limiting  radii  remaining  unchanged. 

TRANSFORMED    PARABOLIC    WHEELS. 

As  an  example  of  a  decided  change  in  the  appearance  of  awheel 
take  those  shown,  in  Fig.  48,  and 
move  the  center  of  motion  from  A  to 
A',  Fig.  57,  thus  adding  the  distance 
A  A'  to  all  the  radii,  the  elementary 
arcs  being  preserved  in  length. 

Divide  the  arc  Cabd  into  equal 
parts  and  draw  the  radial  lines  aAi, 
bAj,  etc. 

Then  the  radius  of  the  point  C  of 
the  transformed  wheel  will  be  A'C, 
and  of  the  point  e  it  is  A'e,  equal  to 
aAi,  where  arc  Ce  =  Ca;  also  radius 
A'f  —  bAj,  and  arc  ef  =  ab,  etc. 

The  arc  hCx  of  the  transformed  wheel,  when  CA'  =  \CA,  will 
be  nearly  a  straight  line,  but  not  exactly;  because  to  be  straight 
requires  the  contour  of  B  to  be  a  logarithmic  curve  resembling  the 
parabola.* 

WHEELS   OF   COMBINED   SECTORS. 

An  interesting  series  of  wheels  may  be  obtained  by  combining 
sectors  of  the  above  various  forms,  as,  for  example,  of  the  ellipse 
and  log-spiral. 

In  Fig.  58  we  have  for  wheel  A  a 
90- degree  elliptic  sector  A  CD  like  the 
fourth  part  of  B,  Fig.  44,  while  the  re- 
maining 270-degree  sector  is  a  log-spiral. 
Wheel  B  is  a  copy  of  A. 

In  Fig.  59  we  have  a  2-lobed  wheel  A, 
each    lobe  of  which    is    composed    of    a 
FIG.  58.  90-degree  elliptic  sector  like  BCT,  Fig.  44, 

and  a  90-degree  log-spiral  sector;  working  with  a  3-lobed  wheel  B, 

*  The  equation  of  the  logarithmic  curve  of  B,  making  the  mating  curve 

JiCx  a  straight  line,  is  y  =  a  log  -  — ,  where  x  and  y  are  co- 

a, 

ordinates  of  the  curve  BC,  and  a  =  the  distance  A'C".     The  polar  equation  of 
the  outline  of  the  wheel  A'  is  r  =  a  sec  v\  v  being  tingle  CA'e,  etc. 


42 


PKINCIPLES   OF   MECHANISM. 


each  lobe  of  which  has  approximately  a  60-degree  elliptic  sector- 
like  B'C'U,  Fig.  44,  and  approximately  a  60-degree  log-spiral  sec- 
tor. If  wheel  A  is  first  made  with  90-degree  sectors  wheel  B  is 
best  laid  out  by  trial  values  of  the  sectoral  angles,  since  the  angles 


FIG.  60. 


FIG.  59. 

will  not  be  exactly  60  degrees  in  B.     When  a  lobe  of  120  degrees  is 

found  by  trial  for  B,  the  remaining  two  lobes  can  be  copied  from  it. 

In  Fig.  60  DFC  and  CGE  are  a  pair  of  rolling  half -ellipses 

with  axes  of  rotation  at  A  and  B. 
The  other  half  of  each  wheel  con- 
sists of  log-spiral  sectors  ADH 
and  AJC,  separated  by  a  circular 
sector  AHJ,  laid  out  by  first  as- 
suming the  circular  sectoral  angle 
CAI  =  HAJ,  then  drawing  in  the 
log-spiral  DHI,  according  to  Figs.  20  and  21,  then  findingt  he 
middle  point  H,  and  transferring  the  sector  A  HI  to  AJC,  when 
the  circle  arc  HJ  completes  the  drawing  of  the  half-wheel  A.  The 
mating  half-wheel  of  B  is  a  copy  of  ADHJC. 

In  Fig.  60  the  wheels,  though  composed  of  several  sectors,  are 
still  similar,  while  in  Fig.  61  we  find  sev- 
eral sectors  combined  in  wheels  that  are 
quite  dissimilar.  Here  CD  and  CG  are  a 
pair  of  elliptic  sectors,  DE,  EF,  arid  GH, 
HI  log-spiral  sectors,  and  CF  and  CI 
elliptic  sectors. 

The  above  examples  will  serve  to  illus- 
trate the  fact  that  a  great  variety  of  wheels 
may  be  made  from  the  four  curves  named  as  the  foundation. 

There  are  many  cases,  however,  where  none  of  the  above  wheels 
will  answer,  as,  for  instance,  where  a  particular  and  definite  law  of 


FIG.  61. 


TRANSFORMED    WHEELS.  4& 

angular  velocity  foreign  to  the  above  curves  must  be  closely  fol- 
lowed during  a  part  of,  or  even  throughout,  the  entire  revolution, 
while  the  above  wheels  have  laws  of  motion  of  their  own  which  will 
rarely  fit  the  special  case  of  a  predetermined  law. 

The  above  wheels  are  fairly  well  adapted  to  the  simple  require- 
ment of  a  specified  maximum  or  minimum  velocity,  or  both,  as  occur- 
ring within  a  revolution,  and  at  certain  points  therein. 


CHAPTER  III. 

NON-CIRCULAR  WHEELS  IN  GENERAL. 

IN  modern  practical  mechanism  there  exists  a  demand  for  non- 
circular  gearing  where  the  law  of  motion  of  the  driven  wheel  is 
fixed  by  considerations  foreign  to  log-spirals,  ellipses,  etc.;  which 
law  the  required  wheels  must  follow,  sometimes  approximately, 
and  sometimes  exactly,  in  some  cases  for  a  part  of,  and  in  other 
cases  throughout  the  entire  revolution  of  the  driven  wheel. 

Again,  there  may  be  demanded  similar  wheels,  differing  from 
log-spirals  and  ellipses,  and  which  can  be  readily  drawn,  and 
readily  changed  and  redrawn  to  suit  the  designer's  fancy.  Hence 
there  arise  three  cases,  viz. : 

I.  One  wheel  given,  or  assumed,  to  find  its  mate  ; 
II.  Laws  of  motion  given  to  find  the  wheels  ;  and 
III.  Similar  wheels. 

In  either  case  the  wheels  may  be  complete  or  incomplete. 
Treating  these  separately,  we  have  : 

I.     GIVEN   ONE  WHEEL   TO   FIND   ITS   MATE. 

In  Fig.  62,  let  A  represent  the  assumed  wheel  so  chosen  that 

when  the  wheel  B  is  found  its  cir- 
cumstances of  motion  will  approx- 
imate those  desired,  more  or  less 
closely  according  to  judgment  in 
assuming  A.  When  B  is  completed 
and  tried,  and  found  not  to  move  as 
desired,  A  may  be  modified  accord- 
ing to  dictates  of  the  first  trial,  and 
FlG-  63-  B  found  again. 

In  drawing  the  wheels  the  outline  Cabc,  etc.,  may  be  traced,  and 
the  center  of  motion,  A,  arbitrarily  assumed. 

44 


NON-CIRCULAR   WHEELS   IK    GENERAL.  45- 

Then  assume  a  point  B  for  the  center  of  motion  for  the  wheel* 
Cmno,  and  draw  the  line  of  centers  AB.  C  is  one  point  in  the 
outline  of  B.  Then  describe  an  arc  am,  produced,  with  C  as  the 
center.  Also  subtract  Aa  from  the  line  of  centers  AB,  and  with 
the  remainder,  equal  Urn,  laid  off  from  B,  find  the  point  m,  as  the 
second  point  in  the  wheel  B.  In  other  words,  with  Aa  as  a  radius, 
and  the  center  A,  describe  the  arc  as.  With  B  as  a  centre  and  Bs 
as  radius,  describe  the  arc  sni,  when  the  intersection  m  is  the  second 
point  in  B.  Next,  with  ab  as  radius  and  m  as  centre,  describe  the 
indefinite  arc  at  n\  and  with  Ab  as  radius,  describe  the  arc  bt,  and 
with  Bt  as  radius,  describe  the  arc  tn,  to  find  the  intersection  n, 
as  a  third  point  in  the  wheel  B. 

Thus  we  have  Cm  =  Ca,  nm  =  ab,  no  =  be,  etc. ;  and  Bm  — 
AB  -  Aa,  Bn  =  AB  -  Ab,  etc. 

"Proceed  thus  for  the  entire  circumference  of  A,  resulting  in 
corresponding  or  mating  points  for  B.  A  curve  traced  through 
these  points  gives  the  first  trial  outline  for  the  wheel  B. 

Now  if,  when  the  center  point  B  is  assumed,  it  makes  the  line 
of  centers  AB  too  great,  the  wheel  B  will  have  a  gap  in  it;  and  con- 
versely if  AB  be  taken  too  small  there  will  be  an  overlap. 

To  make  B  complete,  that  is,  to  just  close  up  without  a  gap  or 
overlap,  a  new  point  for  B  must  be  assumed,  and  the  whole  work 
of  finding  the  points  mno,  etc.,  repeated,  one  or  several  times,  until 
sufficient  closeness  is  obtained,  and  the  outline  of  B  traced,  thus 
completing  the  pitch  lines,  or  wheels,  A  and  B. 

The  distance  to  move  B  will  be  not  far  from  one-sixth  of  the 
arc  of  the  gap,  or  overlap. 

The  above  tentative  process,  as  unscientific  as  it  may  appear, 
is  nevertheless  the  only  way  for  the  case  of  one  wheel  given  to 
find  its  mate. 

As  to  accuracy,  to  make  the  spaces  Cdbc,  etc.,  excessively  great 
introduces  an  error  due  to  the  difference  between  the  chord  and 
arc  ;  while  to  make  them  excessively  small,  the  errors  of  graphical 
work  accumulate.  Probably  10-degree  angles  around  A  will  be 
found  about  right. 

To  increase  the  length  of  the  steps  Ca,  ab,etc.,  without  jeopard- 
izing accuracy,  Rankine  makes  the  ingenious  application  of  the 
following 

Graphic  Rules  for  Equivalence  of  Lines  and  Arcs. 
First.  To  find  a  straight  line  equal  to  a  circle  arc. 


46 


PKItfCIPLES   OF   MECHANISM. 


To  find  the  tangent  line  DP  =  the  arc  DE,  Fig.  63,  take 
D  0  =  \DE,  and  with  0  as  a  center,  describe 
the  circle  arc  EF,  cutting  DF  in  F.  Then 
DF  =  arc  DE  within  less  than  a  tenth  of  one 
per  cent,  up  to  an  angle  of  30  degrees,  and  in- 
creases as  the  fourth  power  of  the  angle. 
FIG.  63.  Second.  To  find  a  circle,  arc  equal  to  a 

straight  line. 

To  find  the  arc  DE  =  line  DF,  Fig.  64,  take  DO  =  %DF,  and 
draw  the  circle  arc  FE  from    0  as  a  center. 
Then  arc  DE  =  line  DF  within  same  limits 
as  first  rule  above. 

The  arc  DE  must  be  tangent  to  the  line 
DF  at  D.  Hence  the  center  of  the  arc  DE 
is  on  DGf  drawn  perpendicular  to  DF  at  D. 


I) 


O 
FIG.  64. 


The  same  rule  and  construction  apply  for  all 
arcs  starting  from  D,  with  centers  on  line  DG, 
as  for  the  dotted  arcs  shown. 

The  above  percentage  of  error  is  true  for  circle  arcs  only;  other 
•curves  give  rise  to  greater  errors. 

To  Draw  a  Tangent  to  any  Curve. 

To  draw  a  tangent  line  to  the  curve  DOE  at  the  point  0. 
First  draw  any  curve  FG,  cutting  the  curve  DOE,  and  also  a 
series  of  straight  lines  as  DI,  He,  ac,  etc. 
Then  lay  off  the  length  DF  from  0  to 
7,  noting  the  point  /;  Oe  from  G  to  H, 
1  noting  the  point  H\  ao  from  b  to  c,  not- 
ing the  point  c,  etc.     Then  trace  a  curve 
through  the  points  Hcdl,  etc.,  as  shown, 
noting  its  intersection  T  with  the  curve 
FIG.  65.  pa      Then  a  straight  line  through  the 

points  T  and  0  will  be  tangent  to  the  curve  DOE  at  0. 

These  rules  will  be  found  useful  in  the  graphic  operations  of 
problems  in  Mechanism. 

Application  of  Rule  Second  to  the  problem  above,  of  one  wheel 
given  to  find  its  mate.  Here  ACDE,  Fig.  66,  is  a  portion  of  wheel 
A,  and  B  the  proposed  center  of  wheel  B,  with  AB  the  line  of 
centers. 

Draw  Cd  tangent  to  the  wheel  A  at  the  point  C.  With  a  as  a 
•center,  taking  Ca  =  %Cd,  draw  the  circle  arc  DdF extended.  Draw 


NON-CIRCULAR  WHEELS  IN   GENERAL.  47 

the  circle  Dm  from  A,  and  the  circle  mF  from  B,  giving  the  inter- 
section F  of  arc  mF  with  arc  DdF.  Then  F  is  a  point  in  the  per- 
iphery of  B,  since  arc  CD  —  line  <7d  =  arc  CF,  of  £,  according  to 
rule  2d  above. 

To  find  another  point  in  B,  draw  the  tangent  De  at  D,  and  the 
radius  ^4Z>  extended,  noting  the  angle  gli.  Then  draw  the  radius 
BF,  and  the  line  Ff  at  the  angle  with  FB  of  ij  —  gh.  Take  the 
points  c  and  6  at  the  J  distance  De  from.  />  and  from  F.  Then 


FIG.  66. 


with  Je  as  a  radius  and  center  #,  draw  the  arc  eE\  also  from  center 
c  the  arc  /#  extended.  Then,  with  a  radius  AB  —  AE  and 
center  at  B,  cut  the  arc  fG  at  6r.  This  latter  point  is  another 
point  in  the  contour  of  wheel  B,  since  arc  DE  =  line  De  =  line 
Ff  =  arc  FG  of  #. 

Thus  proceed  for  the  whole  circumference  of  A;  and  if  the  cir- 
cumference of  B  closes,  as  by  center  B  having  been  rightly  chosen 
in  distance  from  A,  the  points  C,F,G,  etc.,  will  be  correct  points  in 
the  contour  of  B.  But  if  wheel  B  does  not  close,  assume  another 
position  for  center  point  B  and  repeat  the  work  for  a  new  set  of 
points  C,F,G,  etc.  Finally  when  the  wheel  B  closes,  the  points 
C,F,G,  etc.,  will  be  correct  points  in  the  arc  of  B,  though  they  will 
be  too  few  to  admit  of  drawing  in  the  contour  of  B. 

A  sufficient  number  of  points  intermediate  may  then  be  found 
as  by  the  method  of  Fig.  62,  thus  completing  the  wheel  B. 

The  advantage  of  the  wide  steps  of  Fig.  66  is  to  permit  of  find- 
ing the  correct  position  of  the  center  B  with  comparatively  little 
work. 

In  Fig.  67  is  a  photo-process  copy  of  a  pair  of  wheels  in  metal, 


48 


PRINCIPLES    OF    MECHANISM. 


the  pitch  line  of  the  lower  one  of  which  was  first  traced  by  pencil 
free-hand,  and  the  center  point  assumed,  when  the  upper  or  mating 
wheel  was  found  by  the  process  above  described.  The  wheels  are. 
about  6  inches  between  centers. 


FIG.  67. 


FIG.  68. 


Fig.  68  is  a  drawing  copied  from  a  pair  of  wheels  used  on  a 
paper-cutting  machine  from  England  and  exhibited  at  the  World's 
Fair  of  '93.  The  wheels  were  in  cast  iron,  about  16  inches  between 
centers,  in  which  the  smaller  is  simply  an  assumed  circle,  with  its 
axis  chosen  arbitrarily  in  eccentricity,  while  the  larger  wheel  is  2- 
lobed. 

II.    LAWS  OF  MOTION  GIVEN  TO  FIND  THE  WHEELS. 

The  centers  of  a  pair  of  wheels  chosen  to  illustrate  this  case  are- 
taken  at  A  and  B,  Fig.  69,  with  AB  the  line  of  centers,  and  the 
heavy  outlines  CDFH. .  .X,  and  CEGI .  . .  Y,  the  contour  lines  of 
the  completed  half-wheels,  or  at  least  as  much  as  lies  above  the 
extended  line  of  centers. 

The  wheel  A  in  Fig.  69  is  supposed  to  revolve  about  its  center 
of  motion  with  velocity  constant,  so  that  the  "law  of  motion  "  for 
A  is  very  simple,  viz.,  motion  uniform.  A  semi-circle  aceg,  etc.,  is 
drawn  within  the  half-periphery  of  A  and  divided  into  six  equal 
angles  or  sectors  each  of  30  degrees.  For  convenience  suppose  the 
wheel  A  to  make  a  half -revolution  in  six  seconds,  or  any  one  of  the 
six  sectors  in  one  second.  Then  the  law  of  motion  of  A  is  repre- 
sented graphically  by  the  sectors  into  which  it  is  divided,  these 
several  sectors  passing  over  the  line  of  centers  in  equal  times,  say 


NON-CIRCULAR   WHEELS   IN    GENERAL.  49 

one  second  for  each.     These  angles  and  sectors   may  be   called 
velocity  angles  or  sectors,  or  sometimes  auxiliary  angles  or  sectors* 


FIG.  69. 

If  for  B  the  sectors  were  equal  also,  its  motion  would  be  uni- 
form, and  the  resulting  wheels  would  be  circular  and  without 
special  interest.  But  let  the  sectors  of  B  be  varied  in  angle  as 
shown,  any  one  sector  passing  the  line  of  centers  per  second.  Then 
the  law  of  varied  motion  of  B  may  be  represented  graphically  by 


FIG.  70. 

the  varied  sectors,  as  shown.  Hence  the  laws  of  motion  of  the 
wheels  are  represented  by  the  angular  spaces  laid  off  around  the 
centers  A  and  B.  Though  A,  here,  has  equal  angles  and  uniform 
motion,  it  may  readily  have  a  varied  motion  and  a  correspondingly 
varied  series  of  sectors,  as  for  the  case  that  the  driver  A  has  a. 


50  PRINCIPLES    OF    MECHANISM. 

varied  motion.     But  in  practice  the  driver  will  usually  have  uni- 
form motion. 

Next,  suppose  all  the  sectoral  angles  to  have  sectoral  arcs  drawn 
in,  these  arcs  being  all  of  the  same  length,  so  that  the  narrower  the 
sector  the  greater  will  be  its  length.  Thus  the  angles  for  A  being 
all  the  same,  the  arcs  are  all  the  same  distance  from  A  in  a  circle 
arc.  But  for  B  the  arcs  arrange  themselves  at  varied  distances 
from  By  as  shown  in  Fig.  69. 

The  sectors  of  A  and  B  may  be  regarded  as  in  pairs,  so  that  if 
ac  should  roll  against  lij  the  sectors  would  turn  through  their  angles 
in  the  same  time,  and  the  velocity-ratio  would  be  the  inverse  ratio 
of  the  sectoral  radii.  The  same  being  true  of  the  next  pair  of  sec- 
tors rolling  upon  each  other,  it  follows  that  the  velocity-ratio 
would  change  from  the  one  pair  of  sectors  to  the  next,  in  case  the 
radii  differ,  as  in  fact  they  do. 

Thus  for  A  moving  uniformly,  B  would  have  a  varied  motion 
according  to  its  sectoral  angles,  or  according  to  the  stated  law  of 
motion,  since  the  angles  were  laid  out  according  to  that  law. 

Hence  if  these  wheels,  composed  of  stepped  sectors,  could  be 
kept  revolving  with  these  sectoral  arcs  in  rolling  contact  one  after 
another  in  order,  the  law  of  motion,  in  general,  would  be  correct. 
But  this  would  be  thoroughly  impracticable,  since  it  would  require 
the  distance  between  the  centers  A  and  B  to  be  continually  vary- 
ing. This  is  obviated  by  changing  the  radial  lengths  of  the  sectors 
so  that  their  ratio  in  pairs  will  be  the  same  as  before,  and  their  sum 
equal  the  line  of  centers  AB,  while  at  the  same  time  the  modified 
sectoral  arcs  are  merged  into  each  other  in  continuous  lines,  as 
shown  in  the  lines  CDFH. . .  Xand  CEGI. . .  Y. 

Fig.  70  will  serve  best  to  show  how  this  may  be  done,  where  A  and 
B  are  the  centers,  cAe  a  sector  for  wheel  A,  and  kBm  a  sector  for 
wheel  B,  where  are  cde  =  arc  Mm,  these  sectors  with  lettering  and 
construction  lines  being  taken  from  Fig.  69.  Now  to  find  points 
F  and  G  in  the  final  perimetric  arcs  of  wheels  A  and  B,  such  that 
the  velocity-ratio  for  F  and  G  in  driving  contact  shall  be .  the 
same  as  that  for  the  arc  cde  rolling  on  the  arc  klm ;  it  is  only  nec- 
essary that 

length  AF      Ad 
length  BG~  Bl' 

By  graphic  process  this  is  most  readily  done  by  drawing  lines 
from  A  and  B  to  meet  in  some  point  0,  and  parallels  thereto  from 


NON-CIRCULAR   WHEELS   IN    GENERAL.  51 

J^and  L  meeting  in  u,  where  cM  is  an  arc  struck  from  A9  and  kL 
an  arc  struck  from  B,  and  a  line  OuN,  when 


AM  =  Jd  :  ^JV  ;:  BL  =  Bl  :  BN. 

Now  with  radius  J^Y  find  the  intersection  .F  by  drawing  the  arc 
NF\  likewise  with  the  radius  BNfLiid  the  intersection  (7;  t?/'7  be- 
ing drawn  from  the  center  point  A  and  the  middle  of  ce,  and  Gl 
likewise  drawn  with  respect  km  and  B. 

Fig.  70  is  taken  out  of  Fig.  69,  with  like  lettering;  and  by  re- 
ferring to  Fig.  69,  we  find  that  other  points,  as  DHJ,  etc.,  and 
EIK,  etc.,  were  obtained  in  the  same  way.  Through  these  points, 
when  all  are  thus  determined,  the  contours  of  the  completed  half- 
wheels  A  and  B  may  be  drawn  in  with  smooth  outlines. 

In  practice  the  angles  of  the  sectors  should  not  exceed  10  to 
15  degrees  for  suitable  accuarcy  of  results. 

In  the  figure  we  have  drawn  only  half  of  each  wheel  A  and  B, 
but  of  course  the  remaining  halves  may  be  drawn  in  the  same 
way. 

In  practice  these  wheels  have  been  made  where  the  specific  law 
of  motion  of  B  was  carried  in  some  cases  through  180  degrees,  and 
in  others  through  the  whole  360  degrees.  At  the  Centennial  of 
1876  there  was  exhibited,  by  B.  D.  Whitney  of  Winchendon, 
Mass.,  a  certain  barrel-stave  sawing  machine  where  the  carriage 
carrying  the  blank  block  for  staves  was  fed  against  the  "  tub-saw  " 
at  a  uniform  rate  of  advance  and  with  a  quick  return ;  the  move- 
ment of  the  carriage  being  made  by  a  crank  and  pitman  connec- 
tion driven  by  non-circular  gears  of  Fig.  71,  15£  inches  in  diame- 
ter. On  examining  the  marks  of  the  saw  on  sawn  staves  it  was 
found  that  the  forward  feed  was  wonderfully  close  to  a  uniform 
rate  of  advance,  proving  that  the  wheels  must  have  been  care- 
fully worked  out  as  to  laws  of  motion,  even  allowing  for  a  short 
pitman. 

In  this  case  the  wheel  A  made  something  like  a  three-fourths 
turn,  while  B  made  a  half -turn  with  uniform  motion  of  stave  block. 

Fig.  71  is  an  accurate  copy,  the  wheels  serving  as  type,  which, 
with  printer's  ink  and  paper,  gave  the  copy  that  is  photographed 
for  the  figure. 

The  divergence  of  the  teeth  near  the  salient  angles  is  noticeable, 


53  PRINCIPLES   OF   MECHANISM. 

and  suggests  blocking  of  the  gears,  and  yet  they  actually  operated  in 
the  most  satisfactory  manner. 

Also  see  Fig.  120  for  another  example  of  careful  laying  out. 

In  another  example  of  these  wheels  the  pitman  was  in  effect  of 
infinite  length,  giving  uniform  motion  to  a  carriage  forward,  with 


FIG.  71. 


a  quick  return,  the  driver  revolving  uniformly  for  about  a  three- 
fourths  turn,  to  the  half  turn  of  B,  as  required  in  a  boot  and  shoe 
screw  nailing  machine.  See  Figs.  134,  135,  and  136. 

As  a  case  of  quick  return  with  short  pitman  with  a  little  more- 
latitude  for  return,  and  for  starting  and  stopping  at  the  ends  of  the= 
uniform  motion,  see  Fig.  72. 


PIG.  72. 

At  the  Centennial  the  problem  was  made  known  of  a  motion  for 
winding  yarn  on  a  conic  bobbin,  such  that  the  yarn  would  be  taken 
off  a  reel  uniformly  when  winding  upon  the  cone  from  a  base  of 
four  inches  to  a  tip  of  one  inch,  back  and  fourth  on  the  cone  in 
regular  screw  pitch.  This  requires  the  cone  to  revolve  faster  when 
winding  yarn  on  its  tips  than  when  winding  on  its  base,  and  vice 
versa. 

This  problem  has  since  been  worked  out  in  unilobed  non-circu- 


NON-CIRCULAR   WHEELS   IN   GENERAL.  53 

lar  gears,  such  that  while  the  driver  A  revolves  uniformly,  the 
follower  B  will  have  an  acceleration  for  a 
half-revolution  during  the  windings  down 
the  cone,  and  then  a  retardation  for  a  half- 
re  volution  of  B  during  the  winding  up  the 
•cone.  Thus  one  revolution  of  B  takes  place 
while  winding  down  and  back  the  cone.  Now 
if  the  cone  is  to  have  several  convolutions  of 
yarn,  say  ten,  in  winding  down,  and  then  ten  in 
winding  back,  the  cone  must  make  twenty  rev- 
olutions to  one  of  B.  This  relation  can  be 
obtained  by  interposing  circular  gearing  be- 
tween the  cone  and  B,  with  a  ratio  of  in- 
crease of  speed  of  twenty  to  one.  PIG.  73. 

These  non-circular  gears,  as  unilobed  wheels  in  metal,  are  shown 
in  Figs.  73  by  photo-process  copy. 

In  Fig.  89  these  same  wheels  are  constructed  as  bevel  non-cir- 
cular wheels. 

In  Figs.  76  to  78  the  question  of  determination  of  the  pitch 
lines  of  the  non-circular  wheels  for  this  bobbin-winding  problem  is 
fully  solved  as  a  particular  example  in  illustration  of  the  applica- 
tion of  the  principles  of  Figs.  69  and  70;  the  example  answering 
to  the  case  of  laiv  of  motion  definite  and  unalterable  for  the  entire 
revolution. 

Even  this  is  simpler  than  the  most  general  possible  case  where 
the  law  of  motion  for  the  second  half  of  the  revolution  of  the 
driven  wheel  is  different  from  that  of  the  first  half,  as  it  may  be; 
for  which  case  the  resulting  wheels  are  non-symmetrical,  and  may 
resemble  the  wheels  of  Fig.  67,  for  which  wheels,  of  course,  assum- 
ing the  driving  wheel  in  uniform  motion,  the  driven  wheel  follows 
some  law  of  motion,  simple  or  complex,  which  law,  however  com- 
plex, may  be  made  out  and  expressed  graphically,  where,  instead  of 
using  velocity  sectors  as  in  Figs.  74  and  78,  passing  time  may 
represent  the  abscissas,  and  where  the  ordinates  represent  veloci- 
ties of  the  driven  wheel  for  the  corresponding  moments  of  time  or 
•corresponding  points  of  revolution  of  the  driver. 

As  an  example  of  slightly  unsymmetrical  wheels,  suppose  the 
tool  of  Fig.  75  were  required*  to  return  at  a  uniform  rate  of 
motion,  as  well  as  to  advance  according  to  that  law.  That  return 
may  be  in  the  same  time  as  the  advance,  for  which  case,  if  B  is  in 
the  prolongation  of  the  line  SU,  the  wheels  would  be  symmetrical. 


54  PRINCIPLES   OF   MECHANISM. 

But  they  would  not  be  symmetrical  when  taking  J?  above  SU  as 
shown;  and  still  more  unsymmetrical  if  the  return  of  the  tool,  still 
uniform  in  velocity,  were  to  be  in  less  or  more  time  than  the  ad- 
vance. For  this,  according  to  the  description  of  Figs.  74  and  75, 
the  scheme  of  velocity  angles  of  the  lower  half  of  B  is  to  be  made 
in  the  same  way  as  the  upper  half  shown.  Then  A  is  to  be  divided 
into  two  parts,  apportioned  according  to  the  assignments  of  time 
for  the  advance  and  the  return  of  the  tool  T,  those  parts  to  be  di- 
vided into  velocity  sectors  corresponding  in  numbers  with  those  of 
the  mating  parts  of  B. 

SOLUTIONS  OF  SOME  PRACTICAL  PROBLEMS. 
FOR   LAWS   OF   MOVEMENT    GIVEN   TO   FIND   THE    WHEELS. 

As  this  subject  is  evidently  one  of  considerable  importance  in 
practical  mechanism,  the  solution  of  several  problems  is  here  given 
to  illustrate  the  application  of  the  above  principles. 

1st.  The  Shaping-Machine  Problem.— This  is  called  the  shaping- 
machine  problem  because  on  the  machinist's  shaping  machine  the 
cutting  tool  is  desired,  to  have  a  uniform  motion  forward  for  the 
cutting  stroke,  and  a  quick  return  stroke  where  the  tool  returns 
idle  and  should  do  so  quickly  to  save  time. 

For  this  we  have  given  a  shaft,  A,  revolving  uniformly,  gear- 
ing with  a  second  shaft,  B,  revolving  at  such  variable  rate  of  rota- 
tion as  to  secure  the  uniform  forward  and  quick  back  stroke  as 
described;  also  a  carriage  for  the  tool  connected  to  the  driven 
wheel  B  by  pitman  and  crank. 

In  Fig.  74,  T  is  the  tool  to  move  along  the  cut  QR,  held  on  the- 
slide  ME  and  guided  by  NO.  The  slide  EM,  carrying  the  tool,  is 
connected  to  the  crank  BD  by  the  pitman  DE,  E  being  the  "  cross- 
head"  pin  moving  in  the  line  or  path  SU9  which  pin  moves  with 
the  tool. 

Divide  the  path  SUmto  equal  parts  to  represent  uniform  mo- 
tion, the  pin  passing  over  these  parts  forward  in  equal  times.  The 
pitman  DE  =  FS,  =  HU,  =  GV,  etc.,  as  shown.  Hence  the 
crank  pin  D  must  move  through  the  arc  spaces  FG,  GD,  DI,  IJr 
JK,  KH,  in  equal  times,  from  which  we  have  the  angles  FBG,  GBDy 
DEI,  etc.,  to  represent  the  law  of  motion  of  the  driven  wheel  B  for 
the  forward  stroke.  The  back  stroke  being  made  quick,  simply, 
we  may  assume  the  angles  HBL,  DBF,  and  PBF,  which  for  the 
quick  return  should  be  larger  than  those  for  the  forward  stroke. 

Thus,  by  aid  of  Fig.  74,  we  obtain  the  angles  to  be  passed  over 


SOLUTIONS   OF    SOME    PRACTICAL    PROBLEMS. 


55 


by  the  wheel  B  for  its  entire  revolution.  Taking  WXfoi'  the  line 
of  centers  through  A  and  B,  we  may  transfer  the  velocity  angles  to 
Fig.  75  with  like  letters. 


FIG.  74. 


In  Fig.  75,  as  was  done  in  Fig.  69,  draw  equal  arcs  in  the 
angles  about  B,  as  ac  —  de  =  fg,  etc.,  and  extend  radii  from  their 
center  points  as  shown,  nine  in  all.  Taking  A  to  revolve  with 
uniform  velocity,  there  will  be  nine  equal  angles  and  arcs  about 
the  axis  A  as  shown,  these  arcs  being  equal  those  of  B,  viz.: 
mn  =  no,  etc.,  =  ac  =  de,  etc.  Draw  extended  radii  from  the 
centers  of  these  arcs  as  shown.  Then  the  final  radii  for  the  fin- 
ished wheels  may  be  found  by  graphic  proportion,  by  aid  of  the 
triangle  ABO,  as  explained  in  Fig.  70.  Thus  make  Bh  =  B19 
7is  parallel  to  BO,  us  parallel  to  AO,  and  draw  the  line  OsL 
Then  make  Ar  =  At,  and  Bp  =  Bt,  and  we  have  two  points  r  and 


PRINCIPLES   OF   MECHANISM. 


p  in  the  final  peripheries  of  the  wheels  A  and  B.  The  other  points 
:are  found  in  a  similar  manner,  as  explained  in  Fig.  69,  and  the 
iinal  wheels  are  drawn  in  by  tracing  lines  through  the  points  thus 
found. 

The  curves  thus  found,  it  is  to  be  understood,  are  to  serve  as 
the  pitch  lines  of  a  pair  of  gear  wheels.  Figs.  71  and  72  are  ex- 
,amples  of  completed  wheels  by  this  process. 

The  shaper  in  practice  will  usually  have  the  crank  pin  adjust- 
able to  different  radial  distances.  In  this  case  the  above  solution 
should  be  made  for  the  crank  at  some  mean  position,  where  it  is 
likely  to  be  most  used,  as  the  law  will  be  slightly  deviated  from  for 
&  different  position  of  the  pin. 

2d.  A  Bobbin  Winding  Problem. — This  problem,  stated  above, 
of  winding  yarn  on  a  conical  bobbin,  or  "  cop,"  while  feeding  on 
the  yarn  spirally  up  and  down  the  cone,  and  varying  the  speed  of 
revolution  of  cone  so  as  to  draw  the  yarn  at  a  uniform  rate  from 
a  reel  to  avoid  varied  tension  of  yarn  while  winding,  is  believed 
to  possess  sufficient  interest  as  an  application  of  Fig.  69  to  warrant 
its  practical  solution  here. 

Fig.  76  represents  the  conical  bobbin  with  a  spindle-like  exten- 
tension,  showing  also  the  cop  of  yarn  in  section. 
This  is  said  to  be  the  best  form  of  bobbin  and 
cop  from  which  to  supply  yarn  to  knitting 
machines,  since  the  yarn  will  lift  off,  and  up- 
ward, from  the  position  of  Fig.  76,  with  the 
slightest  and  with  the  most  uniform  resistance — 
conditions  of  utmost  importance  to  securing  a 
smooth  knitted  web. 

To  preserve  the  same  cone  form  of  cop  from 
end  to  end,  the  yarn  must  be  fed  upon  the  cone 
in  a  spiral  or  screw  of  uniform  pitch,  from  base  to 
spindle  at  vertex,  and  vice  versa.  The  number  of 
convolutions  winding  up,  and  the  number  back 


u 

FIG.  76. 


again,  is  arbitrary,  and  perhaps  should  be  20  to  40.  But  to  simplify 
the  present  illustrative  example,  take  it  four  ;  that  is,  four  turns  of 
the  bobbin  in  winding  from  base  to  spindle,  and  four  turns  in  wind- 
ing back,  eight  turns  in  all.  This  multiplying  of  the  motion,  8  to  1 , 
may  be  done  by  using  circular  gearing  between  the  non-circular 
gear  wheel  B  and  the  bobbin,  causing  the  bobbin  to  turn  eight 
times  as  fast  as  B.  This  gearing,  as  by  a  screw  of  uniform  pitch, 


SOLUTIONS  OF  SOME  PRACTICAL  PROBLEMS. 


57 


is  to  work  the  feeding  eye  through  which  the  yarn  is  to  pass  in  being 
guided  upon  the  cone. 

Developing  the  cone  with  a  single  layer  of  thread  upon  it,  we  have 
the  sector  ODE,  Fig.  77.  DG  rep- 
resents one  convolution  of  yarn,  KN 
another,  LP  another,  and  1J  the  last. 
These  are  Archimedean  arcs,  and  may 
be  put  in  a  continuous  spiral  by  shift- 
ing the  second  convolution  KN  around 
the  center  0  till  KN  falls  at  GH9  LP 
at  HI,  meeting  //  in  the  case  the  cone 
is  so  assumed,  as  here,  that  DE  is  the 
third  of  the  circle.  The  whole  thread 

winding  down   the   cone   is   then 


in 


FIG.  77. 


DGHIJ,  and  in  winding  back  is  the 
same  taken  in  the  inverse  order. 

Now  divide  this  thread  into  a  certain  number  of  equal  parts, 
DQ,  QR,  R8,  ST,  TU,  UJ,  six  in  all.  These  portions  must  be 
wound  upon  the  bobbin  in  equal  times,  in  order  to  take  the  thread 
uniformly  from  the  reel  as  required.  Therefore  the  angles  DOQ, 


FIG.  78. 

QOR,  ROS,  etc.,  must  be  passed  in  equal  times  by  the  finished 
wheel  B,  while  the  wheel  A  moves  through  equal  angles  in  the  same 
•equal  times. 

In  Fig.  78  lay  off  the  six  angles  hBi,jBk,  IBm,  etc.,  as  shown, 
varying  in  the  same  proportion  as  do  the  angles  DOQ,  QOR,  ROS, 
etc.,  in  Fig.  77;  taking  care  that  the  six  angles  in  Fig.  78,  added 


58 


PRINCIPLES   OF   MECHANISM. 


together,  equal  180°.  For  this  result  we  have  the  added  angles  in 
Fig.  77  equal  360°  +  120°  =  480°;  and  hence  multiply  the  angle 
DOQ  by  Jff  =  -|,  giving  the  angle  UBi,  Fig.  78.  Also  multiply 
angle  QOR,  Fig.  77,  by  f,  giving  the  angle  jBk,  etc. 

Then  draw  in  circle  arcs  hi,  jk,  Im,  etc.,  equaling  correspond- 
ing arcs  in  A,  and  proceed  as  in  Fig.  73  as  shown,  completing  the 
half-wheels  A  and  B,  for  which  all  the  work  is  given  in  Fig.  78. 
The  other  halves  are  the  same,  A  being  symmetrical  with  respect  to 
the  line  AC  as  an  axis  of  symmetry,  and  likewise  for  B. 

These  wheels  are  one-lobed;  A,  the  driver,  differing  somewhat 
from  B  in  shape;  and  as  constructed  carefully  in  metal  by  above 
process,  are  shown  in  Fig.  73  for  the  same  problem. 

Motion  of  Driver  Ay  a  Variable. — In  all  the  above  problems,  the 
velocity  of  the  driver  A  has  been  supposed  to  be  constant,  but  the 
solution  is  readily  made  where  its  velocity  is  variable,  for  which 
case  it  is  only  necessary  to  make  the  series  of  velocity  sectors  un- 
equal to  suit  the  varied  motion,  as  was  done  for  B,  instead  of 
making  them  equal. 

Case  of  Multilobed  Wheels. — In  the  above  wheels,  both  driver 
and  driven  have  been  treated  as  unilobed,  but  multilobes  are  pos- 
sible, particularly  for  the  driver.  For  this,  for  a  bilobed  wheel  A> 
it  is  only  necessary  to  put  into  the  scheme  of  angles  for  A  twice  as. 
many  velocity-angles  as  for  B,  three  times  as  many  for  a  trilobed,, 
etc. ;  and  similarly  for  a  multilobed  wheel  B. 

III.    SIMILAR   WHEELS. 

1st.  Case  of  Similar  and  Equal  Unilobed  WJieels  from  As- 
sumed Auxiliary  Angles. — In  Fig.  79,  A  and  B  are  centers  of  mo- 


tion of  the  wheels.  About  A  draw  a  series  of  auxiliary  sectors,  and 
about  B  the  same  series  of  sectors.  Then,  according  to  Fig.  69, 
draw  equal  arcs  in  all  the  sectors,  and  extended  radii  from  the 
centers  of  the  arcs  as  shown.  Then  the  proportional  radii,  as  of 


SIMILAR   AND    EQUAL   WHEELS.  59 

the  auxiliary  sectors  and  of  the  perimeters,  may  be  found,  when 
the  wheels  may  be  drawn  in  according  to  Fig.  70. 

Fig.  79  shows  the  halves  of  a  pair  of  wheels.  These  may  b& 
copied  for  the  opposite  halves,  when  the  wheels  will  each  be  sym- 
metrical; or  another  set  of  different  auxiliary  sectors  may  be  drawn 
below  AB  and  unsymmetrical  similar  wheels  produced. 

Fig.  129  is  an  exam  pie.  of  a  pair  of  wheels  of  this  kind  in  metal. 

2d.  Similar  and  Equal  Multiloled  Wheels. — In  Fig.  80  we  have 
the  half  of  a  3-lobed  wheel  where  the  lobes  are  all  similar  to 
each  other,  and  one  wheel  similar  and  equal  to  the  other,  but  where 
the  lobes  are  unsymmetrical. 


FIG.  80. 

The  wheels  are  arrived  at  by  assuming  a  system  of  auxiliary  sec- 
tors in  the  60°  angle  CAD,  and  a  like  system  in  the  60°  angle 
CBG.  Determining  the  proportional  radii  between  sectors  and 
perimetric  points  as  before,  and  tracing  in  the  outline  curves  DO 
and  HC\  we  find  them  similar. 

Again,  assuming  a  new  set  of  auxiliary  sectors,  one  like  the- 
other,  in  the  60°  angles  DAK  and  FBH,  determining  proportional 
radii,  and  drawing  in  the  perimetric  carves  DK  and  FHt  we  will 
find  completed  a  third  of  the  wheel  A ;  also  a  third  of  the  wheel  B. 

By  repeating  these  perimetric  curves  around  through  the  remain- 
ing 60°  angles  of  the  wheels,  we  complete  the  pair  of  equal  and 
similar  3-lobed  wheels. 

Thus  equal  and  similar  wheels  of  any  number  of  lobes  can  be 
drawn. 

3d.  Dissimilar  Multiloled  Wheels. — Each  wheel  of  Fig.  80  has 
lobes  that  are  similar  to  each  other,  but  they  are  evidently  not 
necessarily  made  so,  because  each  pair  of  divisional  angles  CAD  and 
CBH  are  here  equal  to  each  other,  but  not  necessarily  equal  to  the- 
next  pair  of  divisional  angles;  and  besides,  the  systems  of  aux- 


€0 


PRINCIPLES   OF  MECHANISM. 


iliary  angles  and  sectors  in  one  pair  of  divisional  angles  are  not 
required  to  be  the  same  as  in  any  other  pair. 

4th.  MuUilobes  of   Unequal   Numbers  of  Lobes. — In  Fig.  81 
"we  have  a  3-lobed  wheel  mating  with  a  2-lobed  one,  where  the 


FIG.  81. 

half-lobe  CAD  of  A  mates  with  the  half-lobe  QBE  of  B\  and 
the  half-lobe  CA  F  with  CBG.  The  sector  DA  PC  is  the  third 
part  of  A,  and  EBGC  the  half  of  B. 

In  a  similar  manner,  A  may  have  any  assumed  number  of  lobes, 
and  likewise  for  B. 

In  Fig.  81  the  angle  of  the  half-lobe  CAD  =  CAP,  and  also 
CBE  =  CBG,  but  they  may  be  unequal,  though  it  would  seem  ad- 
Tisable  for  good  results  that  the  half -lobe  angles  have  the  relation 

CAP       CBG 
CAD  ~  CBE9 


and  that  adjacent  pairs  of  auxiliary  angles  in  adjacent  half-lobes  be 
not  greatly  different  in  ratio  of  radii,  that  the  perimetric  curves 
may  enter  upon  each  other  at  the  limits  of  sectors. 

But  it  is  plain  that  the  auxiliary  angles  in  any  half-lobe  are 


DISSIMILAR   WHEELS.  61 

entirely  arbitrary  except  at  the  lobe  limits  as  above  stated.  In  this 
way  the  auxiliary  angles  of  a  half-lobe  may  be  changed  at  pleasure, 
thus  modifying  the  perimetric  arcs  to  suit  fancy  or  requirements; 
it  being  only  needful  to  note  that  the  half-lobes  that  come  to  mate- 
together  be  related  in  principle,  as  for  the  case  where  the  lobes  in 
one  wheel  are  even  and  in  the  other  odd. 


CHAPTER  IV. 
CASE  II.    AXES  MEETING. 


SPECIAL  BEVEL   NON-CIRCULAR   WHEELS. 

IT  is  usually  advisable  to  refer  wheels  of  this  kind  to  spherical 
surfaces  which  are  normal  to  all  the  elements  of  contact  of  the 
rolling  cones  whose  vertices  are  at  the  center  of  the  sphere  and 
hence  called  normal  spheres',  the  results  of  practical  problems  in 
drawing  these  wheels  being  either  curves  upon  actual  normal 
spheres,  or  ordinates,  angles,  etc.,  by  which  the  curves  may  be  traced 
upon  the  sphere  by  the  pattern-maker  as  he  follows  the  draftsman 
in  the  construction  of  these  wheels. 

Following  the  same  order  here  as  in  plane  wheels,  we  have — 

FIRST.  THE  EQUIANGULAR  SPIRAL. 

1st.  One-Lobed  Wheels.— Take  Fig.  82  to  represent  two  views 
of  the  normal  sphere  on  which  a  pair  of  spheri- 
cal equiangular  spirals  are  drawn  for  a  pair 
of  wheels  A  and  B.  The  outline  of  one  wheel, 
A,  is  shown  at  Cabc  . . ./  and  C'a'b'c' . . ./  in 
the  two  projections;  CBk,  C'B'  being  the 
mate;  a  copy  of  the  first  one,  A  Cf. 

To  draw  this  wheel,  the  actual  normal  sphere 
is  to  serve  best  as  the  drawing-board.  From 
A  A'  draw  a  series  of  meridians  at  equal  inter- 
vening angles  of  not  far  from  5°  each.  In 
Fig,  82  they  are  at  30°  to  avoid  confusion  of 
figure.  We  may  assume  a  point  C,  C',  making 
AC  the  shortest  spherical  radius  of  the  proposed 
wheel.  Then  draw  the  line  Ca,  C'a',  making  a 

certain  angle  with  the  meridian  lying  midway  between  AC  and  Aa. 

From  the  point  a,  a'  where  Ca  intersects  the  meridian  Aa,  draw  a 

62 


AXES   MEETING. 


63 


line  ab,  a'V ,  making  the  same  angle,  on  the  surface  of  the  sphere, 
with  the  meridian  midway  between  A  a  and  Ab  that  the  line  Ca 
did  with  the  meridian  between  A  C  and  Aa.  This  line  intersects 
the  meridian  Ab  at  b.  From  b  draw  the  line  be,  b'c',  making  the 
same  angle  with  the  meridian  midway  between  Ab  and  AC  as  the 
previous  lines  did  with  their  meridians. 

So  continue  till  the  point/,/'  is  reached,  giving  a  half- wheel  of 
one  lobe.  The  points  a' b'c',  etc.,  may  be  symmetrically  copied  upon 
the  other  side  of  C'A'f,  thus  completing  the  drawing  of  a  one-lobed 
equiangular  bevel  non-circular  wheel.  This  wheel  may  be  copied 
at  CBk  for  a  mate,  B,  when  we  have  the  drawing  of  a  pair  of  bevel 
non-circular  wheels. 

These  drawings  may,  however,  have  been  made  upon  normal 
spheres  in  the  form  of  two  separate  spherical  segmental  shells  in 
wood  of  uniform  thickness,  as  shown  in  Fig.  83.     With  drawing 
completed,  the  pattern-maker  could   cut   the 
wood  to  the  finished  wheels  shown,  following 
the  drawing  thus  laid  out  for  him.     The  draw- 
ing, however,  as  above  described,  is  usually  for 
the  "  pitch  line"  of  a  gear  wheel,  and  the  teeth 
may  be  laid  out  before  going  to  the  pattern- 
maker. 

Where  the  mating  wheels,  as  in  this  case, 
are  both  alike  in  pitch  line,  and  perhaps  for 
teeth  as  well,  it  is  only  necessary  to  complete 
one  drawing  and  pattern  to  obtain  castings 
for  any  number  of  pairs  of  cast  wheels. 

One  point  not  to  be  overlooked  is  in  regard 
to  the  size  of  the  wheels  as  above  explained.  They  may  come  out 
so  as  to  require  less  or  more  than  90°  between  the  axes,  and  for 
them  to  come  out  with  any  previously  assumed  angle  of  intersection 
of  axes,  several  trials  of  tracing  the  line  Gabc  .  . ./  may  be  required. 
Any  one  of  these  trial  curves,  however,  will  work  correctly  with 
another  copied  from  itself.  The  angle  between  the  axes  is  the 
same  angle  as  COf,  which  can  be  measured  as  soon  as  the  curve 
Ccf  is  traced. 

This  curve  of  Fig.  82  differs  from  that  of  Fig.  95  of  Mechanical 
Movements,  of  Professor  C.  W.  MacCord,  for  the  reason  that  there 
the  lengths  of  the  meridian-arc  radii  are  made  equal  the  radii  of 
the  plane  equiangular  spiral,  which  process  makes  the  spherical 
spiral  arc  cut  the  meridians  at  a  sharper  angle  for  long  than  for 


FIG.  83. 


64  PRINCIPLES   OF   MECHANISM. 

short  radii.  This  is  most  easily  seen  to  be  true  in  a  quite  eccentric 
1-lobed  wheel,  in  which,  for  short  radii,  the  steepness  of  the 
curve  on  the  sphere  is  nearly  the  same  as  for  the  plane  spiral,  while 
for  the  longer  radii,  nearly  90°  of  meridian  arc,  the  distances  from 
point  to  point  are  much  less  on  the  spherical  than  on  the  plane 
spiral,  while  the  rate  of  change  of  meridian-arc  radii  remains  the 
same  as  for  the  plane  spiral.  It  would  appear  from  this  that  the 
proposed  wheels  of  Professor  MacCord  cannot  work,  because  for 
both  wheels  of  a  pair  the  obliquity  near  the  axis  A  or  B  differs 
from  that  for  points  more  remote;  while,  as  the  wheels  engage  in 
action,  the  shorter  radii, of  one  wheel  mate  with  the  longer  of  the 
other  where  the  obliquity  is  different. 

3d.  Two-Lobed  Wheels,  Multilobed,  etc. — Having  the  curves  of 
Fig.  82  drawn  on  the  actual  normal  sphere,  we  may  take  the  sectoral 
part  C' A'c'  as  the  quarter  of  a  two-lobed  bevel  non-circular  wheel, 
a  pair  of  which  will  work  correctly.  Or  again,  the  sectoral  part 
c'A'f  is  the  quarter  of  a  larger  wheel,  etc.  With  a  spiral  thus 
drawn  on  a  sphere  a  wheel  of  any  number  of  lobes  can  be  made 
out,  any  one  of  which  wheels  will  work  with  another  like  it. 

That  the  axes  of  these  wheels  be  at  right  angles,  the  wheels 
must  be  of  certain  size  and  may  be  made  out  by  trial  measurements 
on  the  spherical  drawing.  This  tentative  process,  though  seem- 
ingly tedious,  is  yet  probably  less  troublesome  than  mathematical 
analysis,  which  in  problems  of  mechanism  often  become  tiresome. 

3d.  Interchangeable  Multilobed  Bevel  Non-Circular  Wheels.— 
Having  drawn  a  spherical  equiangular  spiral  as  in  Fig.  82,  a  sector 
of  180°  may  be  selected  from  it  for  the  half  of  a  1-lobed  wheel. 
A  second  sector  may  be  selected,  of  90°,  for  the  quarter  of  a 
2-lobed  wheel  whose  arc  equals  that  of  the  180°  sector.  Againr 
a  third  sector  may  be  selected,  of  60°, 'for  the  sixth  part  of  a 
3-lobed  wheel  and  whose  arc  equals  that  of  the  180°  sector. 

Any  two  of  these  wheels  will  work  correctly  together. 

The  sphere  will  put  a  final  practical  limit  to  the  number  in  the 
series  of  these  wheels,  for  a  given  spiral.  Another  spiral  will  fur- 
nish another  series  of  interchangeable  wheels. 

Pairs  of  the  above  wheels  selected  to  work  together  will  have 
various  angles  between  the  axes,  the  point  of  intersection  always 
corresponding  with  the  center  0  of  the  sphere  of  Fig.  82. 

Again  sectors  of  different  spirals  may  be  combined  in  a  lobe 
making  unsymmetrical  lobes. 


AXES    MEETING.  65 

SECOND.    ELLIPTIC  BEVEL  NON-CIRCULAR  WHEELS. 

1st.  One-lobed  Wheels. — This  case  is  admirably  treated  by  Mac- 
Cord  in  Mechanical  Movements,  Fig.  88,  where  spherical  ellipses  are 
shown  as  traced  upon  the  normal  sphere  by  means  of  two  pins 
fixed  at  the  focal  points,  around  which  is  placed  a  loop  of  inelastic 
thread,  when  a  pencil,  drawing  the  thread  tight  and  moved  along 
against  the  tense  thread,  traces  the  "spherical  ellipse"  in  a  manner 
entirely  similar  to  that  so  well  known,  of  tracing  the  ellipse  on  a 
plane  surface  with  two  pins  and  a  thread  loop.  Each  ellipse  has 
two  focal  points. 

We  may  take  this  spherical  ellipse  as  the  base,  and  the  center 
of  the  sphere  as  the  vertex  of  an  elliptic  cone,  a  pair  of  which 
will  work  in  true  rolling  contact  when  the 
vertices  are  at  a  common  point,  the  center  of 
the  sphere,  and  the  axes  taken  as  lines  radiat- 
ing from  the  center  of  the  sphere  through  the 
focal  points  above  mentioned,  one  in  each  el- 
lipse, and  at  an  intervening  angle  equal  that 
expressing  the  length  of  the  major  axis  of  the 
elliptic  cone  base;  similarly  as  axes  are  taken  at 
focal  points  A  and  B  in  Fig.  41. 

The  wheels  may  be  finished  to  the  spherical 
form  in  the  same  way  as  illustrated  in  Fig.  83,  the  drawing  being' 
made  on  a  wooden  spherical  blank  or  normal  sphere,  as  drawing- 
board,  as  in  Fig.  84. 

On  a  great-circle  arc  of  the  sphere  assume  the  portion  FADG 
as  the  major  axis  of  the  proposed  spherical  ellipse,  making  FA  = 
D  G,  so  that  A  and  D  may  serve  as  focal  points. 

To  find  a  point  E  in  the  spherical  ellipse,  take  any  arc  FH  in 
the  dividers,  and  A  with  the  focal  point  A  as  a  center,  describe  an 
indefinite  arc  at  E.  Then  with  the  remaining  portion  GH  of  the 
major  axis,  and  with  center  at  the  focal  point  D,  describe  an  arc- 
at  E  cutting  the  former  one,  thus  determining  a  point  E  in  the 
spherical  elliptic  arc.  Find  other  points  in  the  same  way  suffi- 
cient for  drawing  in  the  curve  GEF,  the  half  of  the  elliptic  curve 
required. 

In  order  that  the  elliptic  wheels  may  have  axes  at  right  angles, 
the  angle  FOG  must  be  90°;  but  the  wheels  and  angles  may  be 
greater  or  less  than  this. 


66  PRINCIPLES   OF   MECHANISM. 

Taking  the  line  A  0  as  an  axis,  this  wheel  will  mate  with  an- 
other like  it. 

Instead  of  spherical  wheels  for  this  case,  Professor  MacCord  has 
shown  that  plane  wheels  may  be  obtained  by  cutting  the  elliptic 
cone  by  a  plane  at  right  angles  to  the  line  01,  giving  an  outline  on 
the  plane  which  will  not  be  a  true  ellipse,  though  approximating 
one.  The  contact  point  of  the  pair  of  these  wheels  will  always  be 
found  on  a  fixed  straight  line  connecting  the  axes,  and  perpendicu- 
lar to  that  which  bisects  the  angle  between  the  axes. 

The  advantage  of  the  plane  wheels  over  the  spherical  ones  is 
doubtful,  since  in  laying  out  the  teeth  on  these  blanks  for  gear 
wheels  the  teeth  are  not  normal  to  the  edges  of  the  wheels  except 
at  four  points,  while  for  the  spherical  ones  they  are  normal 
throughout. 

As  in  the  plane  elliptical  wheels  of  Fig.  41,  a  link  may  be 
mounted  upon  pins  fixed  at  the  focal  points  not  occupied  by  the 
axes,  as  at  D  in  Fig.  84  for  wheel  A.  The  pins  at  their  bearings  in 
the  link  must  converge  to  the  center  of  the  sphere. 

2d.  Multilobed  Elliptic  Bevel  Wheels.— These  may  be  treated  by 
applying  the  process  of  Figs.  85,  86,  and  87  to  Fig.  44,  etc. 

A  pair  of  spherical  elliptic  unilobed  wheels  thus  made  have  the 
same  law  of  angular,  velocity  as  plane  elliptic  wheels  of  Fig.  44,  as 
may  be  proved  by  aid  of  Figs.  85  and  86;  while  wheels  of  Fig.  84 
have  not. 

THIRD    AND    FOURTH.     PARABOLIC    AND    HYPERBOLIC    BEVEL 
NON-CIRCULAR  WHEELS. 

These  can  probably  be  best  treated  under  the  general  case  where 
any  pair  of  plane  wheels  can  readily  be  turned  into  bevels.  See 
Figs.  85  to  88. 

FIFTH.     TRANSFORMED  BEVEL  WHEELS. 

Having  given  a  pair  of  bevel  non-circular  wheels,  in  the  form 
as  traced  on  the  normal  spherical  segments,  the  wheels  being  sup- 
posed divided  into  elementary  pairs  of  angles,  arcs,  and  radii,  they 
may  be  transformed  in  the  three  ways  mentioned  for  plane  wheels, 
viz.: 

1st.  By  multiplying  any  pair  of  elementary  angles  by  the  same 
quantity. 

2d.  By  adding  a  constant  length  of  meridian  arc  to  all  the  me- 


AXES   MEETING.  67 

ridional  radii  of  one  wheel,  the  elementary  perimetric  arcs  remain- 
ing unchanged. 

3d.  By  adding  to  one  and  taking  an  equal  length  away  from 
the  other  of  a  pair  of  meridional  radii,  the  perimetric  arcs  remain- 
ing unchanged  in  length. 

Similar  Bevel  Multilobes. — Thus  the  half-wheels  of  Figs.  83  or 
84:  may  be  changed  from  180°  spherical  sectoral  wheels  to  90°  or 
(30°,  etc.,  by  Rule  1st,  just  stated,  and  several  of  them  combined  in 
one  wheel  of  several  lobes,  two  like  ones  of  which  will  work  truly 
together. 

Dissimilar  Interchangeable  Bevel  Multilobes.— By  Rule  2d,  Figs. 
83  and  84  may  be  changed  from  180°  sectors  to  90°,  60°,  etc.,  where 
the  perimetric  arcs,  as  well  as  the  difference  of  the  limiting  spheri- 
cal radii,  will  remain  unchanged  in  length. 

In  this  way  a  2-lobed,  3-lobed,  4-lobed,  etc.,  wheel  may  be  found 
from  Fig.  83,  which  will  mate  with  83or  with  each  other. 

Likewise  for  Fig.  84. 

Partially  Interchangeable  Bevel  Multilobes.— Starting  with  a 
pair  of  correct-working  dissimilar  bevel  non-circular  wheels  as 
those  of  Fig.  78  turned  into  bevels,  one  of  the  pair  may  be  trans- 
formed by  Rule  2d  into  a  series  of  spherical  sectors  of  unchanged 
perimetric  lengths,  and  difference  of  limiting  radii;  but  with  such 
angular  widths  as  to  add  even  to  360°;  and  likewise  treating  the 
other  of  the  primitive  pair,  we  have  two  series  of  wheels,  any 
one  of  one  series  of  which  will  work  truly  with  any  one  of  the 
other  series,  where  no  two  of  either  series  separately  will  work 
together. 

Bevel  Non-Circular  Wheels  of  Combined  Sectors. — Here,  as  in 
plane  wheels,  sectors  of  wheels  of  different  class  may  be  com- 
bined into  one  wheel,  and  the  several  mating  sectors  into  another, 
the  new  wheels  mating  correctly. 

Thus  the  half-wheel  of  Fig.  84  may  be  combined  with  the  half- 
wheel  of  Fig.  83,  the  limiting  radii  being  made  to  agree,  and  both 
constructed  on  equal  normal  spheres;   the  result  being  a  pair  of 
one-lobed  wheels,  one  side  being  elliptic  and  the  other  the  equi 
angular  spiral. 

Again,  each  of  the  above  180°  sectors  may  be  reduced  to  90°  by 
Rule  1st  for  transformation  of  wheels,  or  one  to  70°  and  the  other 
to  110°,  etc.,  and  combined,  making  pairs  of  2-lobed  wheels.  Simi- 
larly, pairs  of  3-lobed,  4-lobed,  etc.,  may  be  brought  out,  but  they 
will  not  be  interchangeable. 


68  PRINCIPLES   OF   MECHANISM. 

Interchangeable  Multilobes  and  Unsymmetrical  Lobes.— Half- 
wheels  of  Figs.  83  and  84  may  be  transformed  by  Rule  2d  into 
sectors- that  may  be  combined,  both  kinds  into  a  single  lobe,  pre- 
serving the  perimetric  arcs  and  limiting  radii,  several  of  which 
lobes  constitute  wheels  such  as  to  realize  interchangeable  multi- 
lobes. 

Velocity- Ratio  in  Bevel  Non-Circular  Wheels.— In  Fig.  7  the 
velocity-ratio  in  circular  bevel  wheels  equals  the  inverse  ratio  of 
the  radii  of  the  wheels,  or  ratio  of  the  perpendiculars  to  the  axes 
from  the  point  of  contact  C.  In  this  way  in  non-circular  wheels 
we  can  find  the  velocity-ratio,  or  number  of  times  faster  one 
wheel  revolves  than  the  other  at  any  instant.  If  the  driver  A 
revolves  uniformly  we  can  find  the  fastest  and  slowest  motion 
for  the  driven  wheel  B\  and  that  one  is  a  certain  number  of  times 
faster  than  the  other  is  often  all  that  is  required,  when  the  above 
wheels  will  answer  the  purpose. 


CHAPTER  V. 
BEVEL  NON-CIRCULAR  WHEELS  IN  GENERAL. 

THE  special  cases  of  bevel  non-circular  wheels  above  considered 
were  those  of  the  equiangular  spiral  and  of  the  elliptic  curve? 
which  could  be  readily  studied  in  the  bevel  form,  and  for  which 
cases  the  law  of  angular  velocity  was  not  important. 

But  in  practical  problems  in  mechanism  where  the  law  of 
motion  of  the  driven  wheel  is  essential,  either  for  a  part  of  or  for 
the  entire  revolution  of  the  driver,  the  forms  of  the  wheels  cannot 
be  assumed  to  be  certain  spirals  nor  ellipses,  but  be  determined  both 
as  mates,  to  suit  the  stated  law  of  motion,  as  was  the  case  in  the 
like  problem  for  plane  wheels. 

As  the  exact  velocity-ratio  depends  upon  perpendiculars  upon 
the  axes  from  the  point  of  contact,  and  not  upon  meridian  circle- 
arc  radii,  a  complication  enters  by  reason  of  which  it  is  found  ad- 
visable to  work  out  the  wheels  as  plane  wheels  first,  and  in  con- 
formity with  the  stated  laws  of  motion,  and  then  to  convert  them 
into  bevels. 

Therefore  the  wheels  are  first  wrought  out,  as  in  Figs.  69,  75,  or 
78,  as  plane  wheels,  after  which  bevel  wheels,  possessing  the  same 
laws  of  motion,  are  determined. 

Take  Fig.  85  for  the  pair  of  plane  wheels  as  worked  out  in 
strict  accordance  with  the  stated  laws  of  motion,  as  in  Fig.  69,  A 
being  the  driver  and  B  the  follower. 

Divide  the  wheels  into  parts  or  sectors  of  5°  or  10°  each,  the 
dividing  lines  serving  as  radii  for  reference. 

In  Fig.  86  draw  the  lines  A  0  and  BO  with  the  same  interven- 
ing angle  A  OB  as  that  to  be  intercepted  by  the  axes  A  and  B  of 
the  finished  bevel  wheels.  Draw  A^B  perpendicular  to  AO, 
where  its  length  is  equal  the  line  of  centers  AB  of  Fig.  85.  Also 
draw  B^A  perpendicular  to  the  line  BO,  where  its  length  is  also 
equal  the  line  of  centers  AB,  Fig.  85.  Then  draw  the  line  AB, 


70 


PRINCIPLES   OF   MECHANISM. 


Fig.  86.  By  this  construction  the  triangle  ABO  is  isosceles.  Also 
draw  the  circle  A'B'  from  0  as  a  center,  to  represent  the  normal 
sphere  in  section. 

b 


FIG.  85. 


Then  lay  off  perpendiculars  AC,  Aa,  Ab,  etc.,  Fig.  86,  equal  to* 
the  radii  AC,  Aa,  Ab,  etc.,  of  wheel  A,  Fig.  85,  and  draw  parallels, 


FA' A,        D      A 
FIG.  86. 


A    A    A 


to  A  0,  as  Cq,  ar,  bs,  etc.,  giving  points  of  intersection  q,  r,  s,  etc., 
on  AB.  Likewise  draw  perpendiculars  BC,  Bm,  Bn,  etc.,  Fig.  86r 
equal  to  the  radii  B  C,  Bm,  Bn,  etc.,  of  wheel  B,  Fig.  85,  and  draw 
parallels  to  B  0,  as  Cg,  mt,  nu,  etc.,  giving  points  of  intersection  q? 
ty  u,  etc.,  on  A  B. 

Then  draw  the  lines  from  the  points  q,  r,  s,  t,  u,  etc.,  to  0,  de- 
termining the  intersection  points  C,  a,  b,  m,  n,  etc.,  on  the  circle 
arc  A'B',  representing  the  spherical  blank  in  section. 

This  completes  the  drawing  preparatory  to  work  on  the  spheri- 
cal blank  in  wood. 


BEVEL   ISTON-CLRCULAR   WHEELS   IX    GENERAL. 


71 


Let  Fig.  87  represent  the  spherical  blank  for  a  pattern  for  the 
wheel  A  in  plan  and  section,  the  radius  AO  being  equal  A' 0  of 
Fig.  86  of  the  normal  sphere.  On  the 
plan  draw  meridian  lines  with  the  same 
scheme  of  intervening  angles  CAa,  aAb, 
bAc,  etc.,  as  in  wheel  A,  Fig.  85.  Then 
with  compasses  opened  to  the  arc  A'C, 
Fig.  86,  place  one  point  at  A,  Fig.  87,  and 
strike  an  arc  at  C,  cutting  the  meridian  at 
the  point  C  as  shown.  Then  the  point  C, 
Fig.  87,  is  one  point  in  the  periphery  of 
the  bevel  wheel  A. 

Another  point  is  found  by  taking  A' a, 
Fig.  86,  in  the  dividers,  and  laying  it  off 
on  the  meridian  Aa  of  Fig.  87,  determin- 
ing the  point  a  of  the  periphery.  Like- 
wise, using  distances  from  Fig.  87,  deter- 
mine points  b,  c,  d,  etc.,  on  all  the 
meridians  of  wheel  blank  A,  Fig.  87, 
when  the  wheel  outline  may  be  drawn  in  and  the  blank  cut  down 
thereto. 

The  sectional  view  of  Fig.  87  shows  how  the  blank  can  be  cut 
away  within  the  outline  to  a  spherical  web,  or  to  the  desired  num- 
ber of  arms,  as  for  the  pattern  from  which  to  obtain  castings  for 
wheel  A. 

A  second  spherical  blank  for  wheel  B  is  also  turned  up  with 
the  radius  AO,  Fig.  87,  equal  A' 0,  Fig.  86,  as  before,  for  wheel  A. 

On  this  blank  are  drawn  meridian  lines  with  the  same  scheme  of 
intervening  angles  as  shown  on  wheel  B,  Fig.  85.  Also  on  all  the 
corresponding  lines  of  the  blank  lay  off  the  arcs  B'm,  B'?i,  etc.,  as- 
in  case  of  wheel  A ;  draw  the  peri  metric  outline,  and  cut  the  wood 
blank  to  shape  as  before,  giving  a  wheel  B  to  mate  with  wheel  A, 
as  a  pair  of  patterns  for  castings  in  pairs,  as  the  final  result  of  the 
problem. 

The  wheel  outlines  thus  found  are  of  course  usually  the  pitch 
lines  for  non-circular  gears,  the  construction  of  teeth  for  which  is 
explained  later. 

In  the  above  the  wheels  were  treated  as  one-lobed.  But  either 
or  both  may  be  of  several  lobes,  or  one  of  a  pair  may  be  internal. 

Usually  the  angle  A  OB,  Fig.  86,  is  a  right  angle,  and  A  B  =  AB 
(of  Fig.  85)  V2. 


72  PRINCIPLES   OF   MECHANISM. 

The  proof  of  the  above  figures  and  process  was  not  attempted 
with  the  explanation.  To  show  it  to  be  the  correct  one  to  carry 
forward  into  the  finished  bevel  wheels  the  same  law  of  angular 
motion  as  the  plane  wheels  of  Fig.  85  possess,  we  note  that  the 
velocity-ratio  for  the  wheels,  as  shown  in  Fig.  85,  is 

Angular  velocity  of  A  _  BC 
Angular  velocity  of  B  ~  A  0" 

In  Fig.  86,  if  OA  and  OB  be  axes,  we  find  that  for  an  arm  A  C 
attached  to  axis  A,  operating  an  arm  BC  attached  to  axis  B,  by 
having  the  ends  joined  together  at  q,  with  the  arms  suitably  located 
on  the  axes,  the  velocity-ratio  will  be 

Angular  velocity  of  OA  _  Eg  _-#£_  GO 
Angular  velocity  of  BO  ~T)q~  AC ~  ~FC 

=  velocity-ratio  of  the  finished  wheels  for  contact  C,  since  FC,  Dg, 
etc.,  are  all  perpendicular  to  the  axes  of  Figs.  86  and  87. 

The  same  is  true  of  all  other  pairs  of  radii.  Hence  the  bevel 
non-circular  wheels  thus  obtained  have  the  same  laws  of  angular 
motion  as  the  plane  wheels  of  Fig.  85,  as  required. 

Fig.  88  is  an  example  in  wood  for  patterns  for  castings,  of  a 


FIG.  88.  FIG.  89. 

pair  of  bevel  elliptic  unilobed  and  bilobed  wheels,  laid  out  by  the 
above  process. 

In  Fig.  89  we  have  a  photo-process  copy  of  a  pair  of  bevel  non- 
circular  wheels  in  metal,  laid  out  by  above  process  from  Fig.  73, 
consequently  possessing  the  same  law  of  angular  velocity.  The 
wheels  are  about  7  inches  between  centers,  and  at  45°  angle  of 
intersection  of  axes. 


BEVEL   NON-CIRCULAR   WHEELS   IN    GENERAL. 


73 


FIG.  90. 


LAYING    OUT    BEVEL    NON-CIRCULAR   WHEELS     DIRECTLY    ON   THE 
NORMAL    SPHERE. 

1st.  One  Wheel  Given,  to  Find  its  Mate.— Let  A  CbD  represent 
the  given  or  assumed  wheel  traced  free  hand  upon  the  normal 
sphere,  the  pair  of  wheels  to  be 
complete  and  not  sectoral. 

Assume  B  for  a  trial  center  for 
wheel  B,  and  the  arc  ACB  on  a 
great  circle  of  the  sphere  for  a  line 
of  centers.  Assume  points  «,  J,  c, 
etc.,  and  find  mating  points  m,  n,  o, 
etc.,  such  that  Cm  =  Ca,  mn  =  ab, 
etc. ;  also  spherical  arc  Bm  -f  arc 
Aa  —  arc  ACB\  arc  Bn  -\-  arc  Ab 
=  arc  ACB,  etc.,  when  the  peri- 
metric  curve  for  wheel  B  may  be 
traced  through  the  points  C,  m,  n, 
o,  etc. 

If  B  was  assumed  too  far  from  A  the  wheel  B  will  have  a  gap 
in  it,  when  another  trial  point  B  may  be  assumed  and  the  work 
repeated  till  the  wheel  B  closes. 

2d.  Laws  of  Motion  Given,  to  Find  the  Wheels.— Let  Fig.  91 
represent  the  actual   normal   sphere,  and  assume   the  wheel  cen- 
ters  A    and    B    on    it   or   axial   lines   A'O 
and  B'O  at  the   desired   intervening  angle, 
say  90°. 

Lay  out  around  A  the  auxiliary  sectors 
with  angles  to  represent  the  motion  of  At 
the  sectoral  angles  being  passed  over  in  the 
movement  of  the  finished  wheel  A  in  equal 
times.  Likewise  draw  the  mating  auxiliary 
sectors  around  B  as  shown. 

To  find  a  pair  of  mating  points  a  and  m 
in   the   peripheries   of   the   finished   wheels, 
make  the  arc  A'h  =  Ae,  and  B'i  =  Bf\  find 
O  k  by  parallels  to  A'O  and  B'O,  and  draw 

FIG.  91.  otym     Then  make  Aa  =  A'j,  and  Bm  =  B'j, 

giving  a  and  m  for  the  desired  mating  points. 

Thus  proceed  with  all  the  auxiliary  sectors,  when  the  wheels 
may  be  drawn  in  through  the  points  a,  b',  m,  n,  etc. 


74  PRINCIPLES   OF   MECHANISM. 

This  will  make  the  velocity-ratio  of  the  finished  wheel  the  same 
as  that  of  the  auxiliary  sectors  rolling  upon  each  other,  because  the 
velocity-ratio  for  the  points  a  and  m  in  contact  at  C  is  the  same  as 
for  the  sector  Ae  rolling  on  the  sector  Bf,  or  the  same  as  for  axis 
A'O  to  drive  axis  B'O  by  contact  of  ends  of  perpendiculars  Ih  and 
ri\  hence 


so  that  the  finished  wheels  have  the  desired  velocity-ratio. 

These  wheels  will  each  turn  through  a  pair  of  mating  angles  in 
equal  times. 

If  A  revolves  uniformly,  as  is  usually  the  case,  the  sectors  for  A 
will  be  equal  to  each  other. 

Similar  Wheels,  Lobed  Wheels,  etc.,  may  be  worked  out  on 
the  normal  sphere,  similarly  as  for  plan  wheels  in  Figs.  79,  80r, 
and  81. 


CHAPTER  VI. 


CASE   m.    AXES 


CROSSING   WITHOUT    MEETING. 
NON-CIRCULAR  WHEELS. 


SKEW -BEVEL 


FOR  this  we  will  attempt  only  the  general  case,  or  the  counter- 
part of  non-circular  bevels  as  brought  out  in  Figs.  85,  86,  and  87, 
as  pertaining  to  non-circular  skew  bevels;  wooden  blanks  being 
here  presumed  also,  as  receiving  part  of  the  work  of  the  draftsman. 

Fig.  92  is  a  pair  of  circular  hyperboloids  answering  for  the  case 


G 


FIG.  92. 


FIG.  93. 


of  pitch  surfaces  for  circular  skew-bevels,  given  to  aid  in  acquaint- 
ing the  mind  with  the  forms  we  have  to  study. 

In  the  present  case  the  rolling  or  pitch  surfaces  are  non-cir- 
cular hyperboloids  with  fixed  axes,  and  a  given  shortest  distance 
between  them. 

75 


76 


PRINCIPLES   OF   MECHANISM. 


At  C  the  surfaces  do  not  coincide,  but  lie  at  an  angle  with  each 
•other  so  that  if  extended  beyond  C  they  would  intersect,  as  in 
Pig.  93  at  F.  This  figure  may  be  taken  to  represent  a  pair  of 
blanks  from  which  we  could  cut  a  pair  of  non-circular  skew-bevels. 
These  should  be  turned  in  a  lathe,  of  nearly  spherical  form,  such 
that  their  line  of  intersection  AFB  as  seen  in  plan  is  a  circular 
line  with  0  as  its  center. 

The  line  AFB,  in  space,  will  be  so  nearly  a  circle  that  it  will 
be  assumed  to  be  one,  and  a  common  meridional  line  of  the  con- 
TCX  surfaces  of  A  and  B,  so  that  DAH  and  EBG  will  be  equal 
spherical  surfaces  intersecting  in  the  line  AFB,  made  to  a  radius 
greater  than  A  0  to  be  determined. 

The  final  result  in  the  laying  out  of  these  wheels  will  be  the 
non-circular  outlines  on  the  blanks  A  and  B,  to  which  the  material 
may  be  cut,  to  realize  the  final  rolling  wheels. 

As  a  first  step  in  the  process  of  obtaining  these  outlines,  deter- 
mine a  set  of  non-circular  wheels  with  axes  parallel,  as  in  Fig.  85, 
that  will  work  in  conformity  with  the  laws  of  motion  as  required 
ior  the  skew-bevels. 


FIG.  94. 

Also  draw  Fig.  86  from  85  as  before,  making  the  angle  A  OB 
equal  the  angle  AOB  of  Fig.  93,  as  the  given  angle  between  the 
axes  in  the  plan  of  the  skew-bevels;  and  the  circle  A'CB',  Fig.  86, 
with  the  radius  AO,  Fig.  93. 

Then  draw  Fig.  94,  where  the  portion  A  OB  is  the  same  as  Fig. 
86,  and  where  OAl  and  OBl  is  the  horizontal  projection  and  KR  and 
O  the  vertical  projection  of  the  axes,  LO  being  the  ground-line. 


NON-CIRCULAR   SKEW   BEVELS.  77' 

In  the  vertical  projection  the  common  perpendicular  between, 
the  axes  is  OR,  projected  in  plan  in  the  point  0. 

The  line  of  contact  between  the  rolling  surfaces  is  here,  in  ele- 
vation, a  shifting  line  PN9  always  parallel  to  KR  and  LO,  and  in 
plan  it  is  a  shifting  radial  line  OD,  PN  and  OD  being  the  projec- 
tions of  that  line  as  in  one  position. 

The  circle  A^DBl  is  taken  as  the  projection  of  the  line  of  inter- 
section AFB  of  the  extended  wheel- bl.*...vs  shown  in  Fig.  93,  A^O* 
being  the  distance  of  the  pole  of  wheel  .1  from  the  common  per- 
pendicular between  the  axes,  B,0  being  the  like  distance  for  wheel 
B\  which  distances  are  equal  A'O  and  B'O. 

The  line  of  intersection  A1DB1  of  the  extended  wheel-blanks  is 
seen  in  elevation  in  the  line  OPK. 

To  correctly  draw  the  line,  divide  the  arc  AlDBl  into  equal 
parts  by  points  i,j,  k,  I,  etc.;  project  lines  up  from  these  points  to 
KR\  then  extend  KR  to  F,  where  the  angle  FOB'=  angle  A^OB^ 
draw  the  semicircle  FeO,  which  intersect  at  e,  f,  g,  etc.,  by  radial 
lines  from  0  to  points  of  equal  division  in  the  arc  B'UVE,  and 
project  lines  from  the  points  e,f,  g,  etc.,  over  to  intersect  the  lines 
previously  projected  up  from  AlBl  to  KR.  Through  the  proper 
points  of  intersection  G,  H,  etc.,  draw  the  line  KP1IGO  for  the 
correct  line  of  intersection  sought.  This  may  be  regarded  as  a  line 
on  a  vertical  circular  cylinder  A^DBl  with  center  at  0,  which,  de- 
veloped, would  show  the  line  OPK  &$  slightly  S-shaped. 

In  the  geometrical  work  this  line  will  be  used  in  its  true  form  ; 
but  for  turning  up  the  wheel-blanks  an  approximating  circle  are 
will  be  used,  found  by  revolving  OPK  around  axis  0  to  the  hori- 
zontal plane  when  the  point  K  describes  the  arc  KL,  falling  at  L,. 
and  Bl  falls  at  /.  To  draw  the  circle  arc  AtJI  &  center  S  is  found 
on  the  line  A^O  extended,  in  such  position  that  A^S  =  IS. 

The  circle  IA^T  is  a  meridian  arc  to  which  the  non-circular 
skew  wheel-blanks  are  to  be  turned  in  the  lathe  in  order  that  their 
line  of  intersections,  as  in  Fig.  93,  may  be  such  as  to  run  through 
AFB,  as  required. 

In  Fig.  95  we  have  the  projection  of  the  wheel-blank  A  turned 
up  to  fit  a  circle  pattern  cut  to  the  circle  IJA^T,  Fig.  94.  On  this 
blank  lay  off  the  angles  CAa,  aAb,  etc.,  as  a  copy  of  the  angles  sim- 
ilarly lettered  on  the  drawing  of  the  plane  wheel,  Fig.  85,  previously 
explained,  drawing  the  radiating  lines  or  meridians  to  the  edge  of 
the  blank,  as  shown  in  Fig.  95. 

To  find  the  point  C  on  the  wheel-blank  radius  A  C,- take  the 
length  Cx,  Fig.  94,  in  the  dividers,  and,  without  changing,  place; 


78 


PRINCIPLES   OF   MECHANISM. 


one  foot  at  P  on  the  curve  OPK  at  such  position  that  the  other 
foot  falls  at  N  on  the  line  OR,  making  PN  perpendicular  to  OR. 

Then,  retaining  the  one  foot  of  the  di- 
viders at  P,  swing  the  dividers,  and  open 
them  till  the  other  foot  falls  at  0.  Retain- 
ing this  length  in  the  dividers,  place  one  foot 
at  /  on  the  circle  A  ^Jl  while  the  other  foot 
falls  on  A^O  at  Y,  making  JY  =  OP  in  the 
diagram  and  perpendicular  to  AtO.  Then 
swing  the  dividers  around  </  and  open  them 
till  the  other  foot  falls  at  the  point  A}.  With 
this  length  A  Jin  the  dividers  lay  off  A  C,  Fig. 
95,  on  the  meridian  A  C  of  the  wheel  blank  A. 
To  find  the  point  a,  Fig.  95,  take  the  dis- 
tance az,  Fig.  94,  in  the  dividers,  and  apply  it 
on  another  line  PN9  P09  JY,  JAl9  as  before, 
and  finally  Aa,  Fig.  95,  and  so  proceed  for 
all  the  points  in  the  periphery  of  A,  when 
the  outline  of  the  wheel  A  may  be  drawn  in. 

Otherwise  explained,  lay  off  Cx9  Fig.  94,  from  R  on  RK.  Pro- 
ject the  end  of  this  line  down  to  P.  Revolve  P  to  Q  about  0. 
Project  Q  to  J.  Then  A1J  =  AO  on  wheel-blank  Fig.  95.  Pro- 
ceed likewise  for  other  points. 

For  wheel  B,  as  for  A  in  Fig.  95,  take  distances  from  C,  m,  n, 

etc.,  to  line  OB',  Fig.  94,  and  apply  these  as  before  from  R  toward 

K.     Project  to  P9  revolve  about  0  to  Q;  project  Q  to  J,  and  take 

Al  to  apply  on  wheel-blank  of  wheel  B  in  the  manner  indicated 

in  Fig.  95  for  meridian  distances  BC,  Bm,  Bn,  etc. 

When  ready  to  cut  the  edge  of  wheels  A  and  B  to  the  perimetric 
lines  just  laid  out,  a  question  will  arise  as  to  what  direction  CD  or 
bevel  to  give  across  the  edge  of  the  wheel,  as  it  will  not  always  be 
normal  to  the  spherical  outer  and  inner  surfaces,  and  the  right-line 
elements  of  the  surface  CD  will  be  askew  with  the  sphere  normals. 
One  way  to  proceed  is  to  cut  the  minimum  non-circular  section 
of  the  wheel  from  a  piece  of  plane  thin  material,  or  thicker  with 
the  edge  bevelled,  as  EF9  Fig.  95,  and  to  mount  this  upon  an  axial 
rod,  FA9  extended  from  the  wheel  blank  A,  Fig.  95,  making 
AF  =  OA,,  Fig.  94,  or  UA9  Fig.  93. 

To  lay  out  the  small  wheel  EF  first  draw  a  series  of  diametric 
lines  with  the  intervening  angles  the  same  as  in  the  upper  part  of 
Fig.  95,  or  in  the  plane  wheels  of  Fig.  85.  The  lengths  of  the 


NON-CIRCULAR   SKEW    BEVELS.  79 

radii  will  be  the  several  heights  ON,  Fig.  94,  which  may  be  noted 
as  the  lengths  Cx,  az,  etc.,  are  placed  in  the  several  positions  PN 
ns  explained.  These  heights  ON  are  to  be  laid  off  on  the  proper 
radii  of  EF,  Fig.  95,  in  order;  when  the  outline  may  be  traced  in, 
and  the  section  EF  cut  to  shape.  The  radial  lines  on  EF,  as  well 
as  on  A,  should  be  plainly  marked  and  lettered,  or  numbered,  to 
distinguish  them.  In  mounting  EFupou  the  rod  the  mean  radius 
approximately  should  be  selected  and  noted,  as,  for  instance,  that 
corresponding  with  AC  in  the  figures,  and  this  radius  on  EF 
placed  at  an  angle  with  the  corresponding  radius  on  wheel,  which 
angle  is  PON,  Fig.  94. 

Then  in  dressing  off  the  edge  of  wheel  A,  Fig.  95,  use  a  thin 
straight-edge,  or  thread  across  from  EF  to  A  C,  trained  upon  cor- 
responding points  of  the  two,  cutting  through  from  C  to  D  till  it 
coincides  with  the  straightened  thread  stretched  from  E  to  C,  and 
likewise  for  the  several  points  of  radii  noted,  when  the  intermediate 
portions  can  be  cut  by  eye. 

Wheel  B  is  to  be  treated  in  a  similar  manner. 

This  gives  us  the  rolling  or  pitch  surfaces  of  the  wheels. 

Instead  of  using  the  small  wheel  EF,  Fig.  84,  as  a  guide  in 
stretching  the  thread  to  use  while  trimming  off  the  material  at  CD 
and  other  points,  it  may  be  advisable  to  use  a  second  blank  like 
AC,  turned  to  the  same  spherical  surface  except  it  may  be  thinner 
by  cutting  away  on  the  concave  side.  Let  us  call  this  blank  M. 

On  the  spherical  convex  side  of  this  second  blank  M  we  are  to 
lay  out  meridians,  straight  or  slightly  curved,  as  explained  for  Fig. 
95,  but  with  intervening  angles  in  the  inverse  order  of  those  of 
Fig.  84.  On  these  meridians  the  same  radial  distances  A  C,  Aa, 
-etc.,  are  to  be  laid  off  also  in  the  inverse  order. 

This  wheel,  cut  to  line  with  quite  a  relief  bevel,  is  to  be  mounted 
upon  an  axial  rod  like  AF,  Fig.  95,  extended  to  twice  the  distance 
AF9  or  to  twice  OAlt  Fig.  94;  and  at  the  intervening  twist  angle 
between  corresponding  radii  of  twice  PON,  Fig.  94;  and  with  the 
concave  sides  of  the  blanks  toward  each  other,  and  twisted  in  the 
right  direction.  See  Fig.  139.  Then  in  dressing  off  the  blank  at 
and  along  CD  attach  the  line  to  the  opposite  blank  M,  at  the  cor- 
responding point  C,  and  draw  the  line  straight,  and  apply  to  CD  as 
often  as  desired  while  dressing  away  the  material,  until  coincidence 
is  obtained  entirely  across  CD,  making  this  one  right-lined  element 
of  the  surface.  If  we  call  this  element  CC9  because  connecting  the 


80  PRINCIPLES   OF   MECHANISM. 

points  C  and  C  of  the  blanks,  the  next  point  may  be  aa,  and  the 
next  bb,  etc.  See  Fig.  95. 

After  dressing  the  blank  across  at  CD,  move  the  line  and  dress- 
ing action  to  the  next  point  aa,  finishing  which  as  before;  move 
again  to  bb,  etc.,  for  the  whole  periphery  of  A.  On  the  portions 
of  surface  of  the  edge  of  A  between  the  line  elements  CD,  etc.,  as 
above  found,  the  material  may  be  cut  away  by  eye,  or  if  preferred 
the  line  may  be  stretched  to  intermediate  points. 

Proof  for  the  construction  of  the  line  OPKy  Fig.  94,  as  de- 
scribed, is  found  in  the  formula  expressing  the  proper  division  of 
the  common  perpendicular  OR,  Fig.  94,  or  A'B',  Fig.  10,  into 
segments  or  gorge  radii,  in  accordance  with  the  velocity-ratio,  viz., 
Fig.  10,  calling  angle  A'C'O'  =  a,  B'C'O'  =  ft,  A'C'B'  =  «,  and 
the  common  perpendicular  =  c. 

O'C'  tan  a  =  A'  0'  =  x\     O'C'  tan  /3=£'0'  =  y; 

,  tan  a      x 

whence  -  -  -5  =  -  ;  also  x  -\-  y  —  c    and     a  -\-  ft  =  a. 

-rn-      •       A-  j    /?  x  *an   (a  ~  a}         C  —  X 

Eliminating  y  and  p,  we  get  —  -  =  -    —  , 

t/an  of  x 

,  .  ,  ,         ,        ,  ,  sin  CK         . 

which  may  be  reduced  to  x  =  c—  --  cos  (a  —  a), 

sin  a 

in  which,  for  Fig.  94,  c  =  OR,  a  =  AOB  —  A1OBl  =  BOE> 
a  =  any  angle,  as  B'OV,  and  x  —  the  corresponding  height  HZ. 

Thus  OF  =  OR  secant  ROF=c.  cosec  a  =  - 


sin  a 


Of  =  OF  cos  VOE  =  --  cos  (a  -  a); 
sin  a 

HZ  =  OP  sin  B'OV=  -r^—  cos  (a  —  a)  sin  a  =  x. 

sin  « 

Hence  it  appears  that  the  height  HZ,  Fig.  94,  is  the  result  of 
the  graphic  construction  of  the  above  formula,  and  that  the  posi- 
tion of  the  point  of  contact  or  intersection  of  the  line  of  contact 
with  the  line  OP  is  correctly  determined  for  each  set  of  radii  of 
the  plane  wheels,  and  that  consequently  the  radii  of  the  wheel 
blanks,  made  out  as  described,  are  correct. 

To  be  strictly  correct,  instead  of  meridians  A  C,  Aa,  Ab,  etc.,  Fig. 
95,  as  mentioned,  those  lines  as  seen  in  projection  should  belaid  off 
as  copies  of  the  line  OPK  as  seen  in  projection  in  Fig.  94,  and  at 
intervening  angles,  the  same  as  before.  Then  FE,  Fig.  95,  should 
be  located  with  respect  to  a  particular  point  P  and  angle  PON. 


CHAPTER  VII. 
NON-CIRCULAR  WHEELS  FOR  INTERMITTENT  MOTION. 

THESE  wheels  might  properly  have  been  considered  under  the 
cases  of  axes  parallel,  axes  meeting,  and  axes  not  parallel  and  not 
meeting,  but  as  their  peculiarities  in  other  respects  are  greater,  it 
was  thought  advisable  to  treat  them  together,  in  deference  to  that 
refinement  of  classification. 

They  are  here  classified  in  two  ways,  with  reference  to  the  de- 
vices for  stopping  and  starting  the  driven  wheel,  viz.: 

1st.  Where  easement  spurs  are  used  together  with  locking  arc- 
of  greater  radius  than  that  of  the  rolling  arc;  and 

3d.  Where  the  locking  arc  is  smaller  than  the  rolling  arc,  with 
easements  upon  the  ends  of  the  pitch  lines. 

1st.  Where  Easement  Spurs  are  Employed. 

In  Fig.  96  we  have  an  example  of  the  first  kind,  where  ease- 
ment spurs  or  prongs  are  used  for  stopping  and  starting  the 
driven  wheel  with  reduced  shocks.  Here  the  arc 
DEF  is  non-circular  and  rolls  correctly  on  the 
arc  GMH,  the  arc  from  near  D  and  F  through  G 
and  H  being  circular  and  serving  simply  as  a  lock 
for  wheel  B  while  standing  still  during  its  period 
of  rest  or  intermitted  movement. 

The  curved  piece  IJ  fast  upon  A,  and  KL  fast 
upon  B,  have  initial  contact  at  the  points  /and  K\ 
when  A  is  rotated  so  that  /approaches  K,  and  when 
/is  carried  down  near  to  At  the  quickness  with  which  B  is  started 
is  reduced,  as  well  as  the  shock  due  to  jt.  Hence  for  rapid  mo- 
tions AT  should  be  short,  while  for  slower  speeds  it  may  be  greater, 
In  some  cases  IJ  may  be  a  mere  pin,  as  in  Fig.  11  for  circular  roll- 
ing curves. 

In  either  case  the  shape  of  DG  and  saddle  curve  GH  should  work 
with  but  little  backlash  when  //and  KL  are  in  running  contact. 

The  spurs  here,  as  well  as  in  Fig.  12,  may  work  by  rolling; 

81 


PRINCIPLES   OF    MECHANISM. 


or  sliding  contact,  though  the  latter  will  be  found  most  convenient 
to  lay  out,  and  no  more  objectionable,  unless  it 
be  in  the  matter  of  lubrication. 


2d.  Where  Locking  Arcs  and  Easements  are 

Employed. 

In  Fig.  97^  the  wheels  have  the  locking  arc 
of  smaller  radius  than  the  rolling  arc  DEF,  in- 
stead of  larger  as  in  Fig.  96.  The  shapes  of  D 
and  F  will  be  much  the  same  as  in  Fig.  15. 

The  rolling  arcs  in  Fig.  97,  as  well  as  in  Fig. 
96,  may  have  any  non-circular  form  desired,  ac- 
cording to  requirement  of  practical  problems. 
As  in  circular  wheels,  these  may  be  made  with  axes  meeting  or 
axes  crossing  and  not  meeting,  according  to  principles  laid  down  in 
the  proper  places  as  pertaining  to  these  problems. 

Fig.  98  is  a  photo-process  copy  of  a  pair  of  wheels  like  Fig.  96, 
turned  into  bevels,  with  axes  meeting  at  60°,  7  inches  between 
centers,  and  of  very  considerable  eccentricity,  as  best  shown  in  the 
second  part  of  Fig.  98. 


FIG.  97. 


FIG.  98. 

The  process  of  Figs.  85  to  87  was  followed  in  laying  out  these 
wheels. 

DIRECTIONAL  RELATION  CHANGING.     VELOCITY-RATIO 

CONSTANT  OR  VARYING. 
ALTERNATE    MOTIONS    BY    ROLLING    CONTACT. 

Pitch  Lines  for  Alternate-Motion  Gearing. 
Among  these  movements  there  are  those  that  are  necessarily 
limited  in  extent  of  movement,  and  others  not. 


NON-CIRCULAR   WHEELS    FOR   INTERMITTENT   MOTION. 


83 


FIRST.  LIMITED  ALTERNATE  MOTIONS. 

Velocity-ratio  Constant. 

In  Fig.  99  the  driving  half-wheel  A  turns  continuously  in  one 
direction  while  the  driven  piece  B  reciprocates,  and  hence  the 
change  in  directional  relation. 


FIG.  99. 

In  the  present  case  the  half-wheel  is  circular,  and  rolls  on  the 
straight  line  DE  for  movement  in  the  one  direction,  the  line  DCE 
being  equal  the  sernicircu inference  HCL  The  arrangement  is  to 
be  such  that  as  the  driven  piece  B,  in  moving  to  the  right,  reaches 
that  position  where  H  comes  to  contact  with  the  extremity  D,  the 
other  end  /  of  the  half-wheel  arc  is  just  on  the  point  of  taking 
contact  at  G,  so  that  by  further  motion  of  A  the  rolling  is  trans- 
ferred from  DE  to  FG  and  so  continues  with  B  moving  to  the  left, 
till  the  last  point,  H,  of  the  wheel  makes  contact  at  F,  when  the 
contact  is  retransf erred  to  DE,  and  the  motion  again  reversed. 

It  is  plain  that  the  greatest  extent  of  movement  of  B  is  the 
length  DE  —  GF  =  HOI,  and  hence  the  movement  is  termed 
"limited." 

Also  it  is  plain  that  if  A  revolves  uniformly,  the  motion  of  B 
will  be  uniform  and  the  velocity-ratio  constant. 

Fig.  99  is  simply  the  theoretical  representation  of  the  move- 
ment. In  practice,  gear  teeth,  as  well  as  suitable  positive  engag- 
ing and  disengaging  devices,  are  to  be  applied,  by  which  the  move- 
ment will  be  still  further  limited,  which  gearing  and  devices  will 
be  treated  under  the  proper  headings. 

Velocity-ratio  Varying. 

In  Fig.  100  the  driver  A  is  a  180°  sector  of  a  log.  spiral  and, 
as  before,  rotates  continually  in  the 
same  direction,  while  the  point  of 
contact  shifts  over  from  the  straight 
inclined  line  DCE  to  the  parallel  line 
FG,  simultaneously  with  which  the 
direction  of  motion  of  B  is  reversed. 

If  A  revolves  with  uniform  velocity,  the  movement  of  B  is 


84 


PRINCIPLES   OF   MECHANISM. 


hastened  as  the  contact  at  C  approaches  D  or  Fy  so  that  the  ve- 
locity-ratio is  a  varying  quantity. 

Drawing  a  line  from  the  center  of  A  through  C  in  a  direction 
perpendicular  to  the  slide  bearings  BB,  we  have  the  line  of  cen- 
ters, the  center  of  motion  of  B  being  at  an  infinite  distance  from 
A,  since  the  line  DC  is  an  arc  of  a  log.  spiral  with  the  pole  at 
infinity. 

The  velocity-ratio  is  the  same  as  given  in  connection  with  Fig. 
27,  the  rate  of  motion  of  B  being  the  same  as  that  of  the  point  C 

as   if    revolving   about  A    with   the- 
angular  velocity  of  A. 

Another  example  of  velocity-ratio' 
varying  is  given  in  Fig.  101,  where 
A  may  be  taken  as  a  portion  of  any 
one-lobed  wheel,  and  the  curve  DE 
FlG-  10L  or  FG  found  such  as  will  mate  with 

it.     The    same  formula  for  velocity-ratio  applies   as  in  Fig.  100' 
or  27. 

SECOND.  UNLIMITED  ALTERNATE  MOTIONS. 
Velocity-ratio  Constant. 

The  Mangle  Rack. 

In  Fig.  102  A  acts  by  rolling  contact  against  the  bar  BB 
secured  to  the  flat  surface  of  DE,  in  which  latter  a  groove,  FG,  is 


o 

1 

-   K 

jr/^"~ 

—     .,                 -           \P 

D 

(  (2 

C                      Bj)    ) 

V 

/"7^\                     ^/ 

~^S/ 

0 

-1    k 

H 

FIG.  102. 

cut  into  which  the  end  of  the  shaft  of  A  is  inserted,  that  A  may 
be  held  in  rolling  contact  with  the  barlike  raised  part  BB.  The 
ends  of  BB  are  rounded,  equidistant  from  which  the  groove  FG 
follows  in  its  circuit  about  BB.  The  piece  DE  is  fitted  to  slide  in 
straight  guides,  so  that  as  wheel  A  continues  to  revolve  in  the 
same  direction,  the  bar  BB  and  attached  slide  DE  will  be  moved 


NON-CIRCULAR   WHEELS   FOR   INTERMITTENT   MOTION. 


85 


till  A  reaches  the  extremity  of  B,  when  further  rotation  of  A  will 
cause  A  to  pass  around  upward  at  one  end  and  downward  at  the 
other  end,  in  continued  revolving  of  A  and  reciprocation  of  BB 
and  DE. 

A  bar  HK  has  a  slot  through  which  the  shaft  of  A  passes  to 
prevent  it  from  swinging  to  the  right  or  left. 

This  mangle-rack  movement  may  be  given  a  piece  BB  of  any 
length,  without  limit.  The  part  BB  may  be  narrow,  even  reduced 
to  a  mere  line. 

The  velocity-ratio  is  the  same  as  that  in  Fig.  27  or  100. 

The  Mangle   Wheel 

In  Fig.  103  we  have  much  the  same 
•construction  as  in  Fig.  102,  except 
here  BB  and  FG  are  formed  in  a 
circle  around  0,  the  part  DE  being 
here  a  circular  disk  mounted  on  a 
center  at  0,  so  that  the  driven  part 
BB  and  DE  has  an  alternating  cir- 
cular motion. 

The  velocity-ratio  is  constant  while 
B  is  moving  in  one  direction,  where 
BB  is  circular  about  0  also  in  the 
other,  though  in  the  second  case  it 
differs  from  that  of  the  first  when  BB 
has  a  sensible  radial  thickness. 

Velocity-ratio  Variable. 

The  velocity-ratio  will  be  variable 
when  the  piece  BB  in  Fig.  102  is 
curved  on  a  straight  slide,  or  straight 
on  a  curved  slide,  or  with  varied 
width. 

For  Fig.  103  the  piece  BB  may  be  of  varied  width,  or  non- 
circular  with  respect  to  the  centre  0,  or  in  form  of  a  spiral,  even 
reaching  to  several  turns  about  0,  and  with  A  non-circular,  com- 
bined with  the  above  modifications. 


JIIG 


PRINCIPLES   OF   MECHANISM. 


AXES   MEETING. 

Bevel  wheels  in  these  movements  are  practical,  and  probably 
skew-bevels  also,  the  construction  of  which  must  have  due  regard 

to  the  principles  of  bevels,  or 
skew-bevels,  as  laid  down  under 
rolling  contact. 

In  Fig.  104  is  given  a  photo- 
process  copy  from  a  pair  of 
mangle  wheels  under  Axes  Meet- 
ing at  a  right  angle.  The  pitch 
line  is  merely  a  circle  on  a  cylin- 
der set  with  teeth,  which  are  sim- 
ply a  row  of  pins  with  which  the 
driver  engages  on  either  side. 


PART  II 
TRANSMISSION  OF  MOTION  BY  SLIDING  CONTACT. 


CHAPTER  VIII. 
SLIDING  CONTACT  IN  GENERAL. 

IN  elementary  combinations  of  mechanism  acting  by  sliding 
contact,  there  are  a  driver  and  a  follower  with  axes  at  a  fixed  dis- 
tance apart,  the  driver  acting  upon  the  follower  at  a  point  of 
contact  at  some  distance  to  one  side  of  the  line  of  centers,  in  con- 
sequence of  which  there  will  be  sliding  of  one  piece  against  the 
other  at  the  contact  point,  since  rolling  action  occurs  only  when 
the  contact  is  on  the  line  of  centers. 

In  thick  pieces  the  contact  will  be  along  a  line  of  tangency, 
which  line  must  be  continually  parallel  to  the  axes  when  the  latter 
are  parallel,  or  meet  with  them  at  a  common  point  when  they  are 
meeting.  In  each  of  the  two  pieces,  all  sections  normal  to  the 
axis  are  similar  figures. 

Velocity-ratio  in  Sliding  Contact. 

In  Fig.  105,  take  A  as  the  center  of  rotation  of  a  piece  AJDK, 
with  contact  at  D  against  the  piece  BLDM.  Then  when  the  piece 
AD  is  turned  through  some  angle  toward  BD,  the  latter  must  turn 
also,  when  a  sliding  action  will  occur  at  the  contact  D.  Hence 
AD  may  drive  BD  by  sliding  contact  at  D. 

To  find  the  velocity-ratio  for  this  turning  of  the  pieces  AD  and 
BD  about  their  centers  A  and  B,  while  thus  transmitting  motion 
by  sliding  contact,  find  the  center  of  curvature  of  the  piece  A  JDK 

87 


88 


PRINCIPLES   OF   MECHANISM. 


or  of  the  curve  JDK  at  D,  as  some  point  E.     Also  find  the  center 
of  curvature  F,  of  the  curve  LDM  at  D,  for  the  piece  BLDM. 

Then  DE  will  be  the  radius  of  curvature  at  D  of  the  curve 
JDK,  and  DFihe  radius  of  curvature  at  D  of  the  curve  LDM ; 
and  pieces  might  be  cut  from  thin  wood,  one  of  shape  AJDK, 
and  the  other  of  shape  BLDM,  which,  when  held  by  pins  at  A  and 
B  for  axes,  could  turn  by  sliding  upon  each  other  in  the  manner 
proposed.  For  a  small  amount  of  movement,  sufficient  to  deter- 
mine the  velocity-ratio,  the  distance  EF  will  be  constant,  it  being 
the  sum  of  the  radii  of  the  curves  in  contact  at  D.  We  observe 
that  in  this  case  E  is  a  fixed  point  on  the  piece  AD,  and  F  a  fixed 


FIG.  105. 

point  on  the  piece  BD.  Now  cut  another  pair  of  pieces,  AFCG 
and  BHCI,  making  E  the  center  of  curvature  of  the  curve  FCG 
at  C,  and  F  the  center  of  curvature  of  the  curve  HC1  at  C,  the 
point  C  being  in  this  case  chosen  where  the  line  EF  intersects  the 
line  of  centers  AB. 

If  now  the  two  pieces  A  EC  and  A  ED  are  fastened  together 
with  a  common  axis  at  A  and  coincident  at  E,  likewise  if  the 
pieces  BFC  and  BFD  be  united  with  a  common  axis  B  and  coin- 
cident at  F,  and  the  two  be  mated  on  their  centers  A  and  B  by 
axial  pins  and  brought  to  running  contact,  then  for  a  small  move- 
ment the  pieces  will  be  in  rolling  contact  at  C,  as  in  Fig.  2,  be- 
cause this  point  is  on  the  line  of  centers;  and  in  sliding  contact 
at  D,  as  at  first  proposed;  and  the  two  actions  will  have  a  common 

7?  C* 

velocity-ratio  which  equals  -r'   as  in  Fig.  2. 

A  Lf 

Therefore,,  to  determine  the  velocity-ratio  of  a  pair  of  pieces 
AJDK  and  BLDM  transmitting  motion  by  sliding  contact,  draw 
a  common  normal  DC  to  the  curves  in  contact,  extending  it  to 


SLIDING   CONTACT   IN    GENERAL.  89 

intersect  the  line  of  centers;  then  the  velocity-ratio  is  given  by  the 

equality 


v  ~  A  6" 

where  Fis  the  angular  velocity  of  A9  and  v  the  angular  velocity  of 
1).  Hence  the  velocity-ratio  in  sliding  contact  is  the  inverse 
ratio  of  the  segment's  ^f  the  line  of  centers  as  the  latter  is  cut  by 
the  common  normal  to  the  sliding  curves. 

In  Fig.  105  only  a  small  movement  was  supposed  to  occur, 
beeause  the  curves  in  common  tangency  at  D  and  C  may  not  be 
circular  to  any  considerable  extent.  But  to  realize  an  indefinite 
movement  under  like  conditions,  it  is  only  necessary  to  suppose  the 
curves  to  be  so  shaped  that  the  point  of  tangency  C  shall  remain 
on  the  line  of  centers  as  for  the  non-circular  wheels  of  Fig.  51, 
while  the  common  normal  EDF  shall  continue  to  pass  through  the 
same  tangency  point  C.  For  this,  the  centers  of  curvature  E  and 
F  may  be  continually  changing  positions. 

The  curves  FCG  and  HC1  may  be  circles  to  the  centers  A  and 
B,  as  in  the  pitch  lines  of  circular  gearing,  when  the  point  C  of 
the  common  normal  CD  becomes  stationary  on  AB. 

SLIDING   AND    ROLLING    CURVES   WITH    COMMON   LAW   OF   MOVE- 
MENT  NOT    REQUIRED   TO    BE   CONCENTRIC. 

In  Fig.  106  let  the  curves  AaCc  and  BbCd  be  a  pair  of  correct- 
working  non-circular  rolling  wheels  in  rolling  contact  at  C  ',  and 


FIG.  106. 


ADe  and  BDfa,  pair  of  pieces  in  action  by  sliding  contact  at  D, 
and  of  such  shape  that  the  common  normal  at  D,  prolonged,  shall 
continually  meet  the  point  C.  Then,  because  the  velocity-  ratio  of 
the  non-circular  wheels  is  B  (7  over  AC,  and  because  the  velocity- 


OF   THB 

UNIVERSITY 


90 


PRINCIPLES   OF   MECHANISM. 


ratio  of  tke  sliding  curves  in  contact  at  D  is  also  BC  over  AC,  it 
follows  that  .Z)  (7  must  be  a  common  normal  at  D,  but  not  neces- 
sarily at  C ;  and  that  the  law  of  the  velocity-ratio  is  common  to 
both  actions  so  long  as  the  curves  are  so  shaped  that  DCis  a  com- 
mon normal  at  D  and  meets  the  point  C. 

Fig.  107  is  a  photo-process  copy  of  a  model  in  metal  of  what  is  • 
shown  in  Fig.  106.     The  lower  sectors  are  the  rolling  arcs,  and  the 


FIG.  107. 


upper  the  sliding  arcs.  Their  lengths  are  such  that  they  all  begin 
and  end  action  together.  The  sliding  curves  cross  the  rolling  curves 
as  in  Fig.  108. 

SLIDING   CUKVES  TO  WORK   CONFORMABLY  WITH   ROLLING  CURVES. 

In  Fig.  108  let  AlCjck  and  BoCmgn  represent  a  pair,  or  por- 


FIG.  108. 

tions  of  a  pair,  of  correct-working  rolling  curves  centered  at  A  and 
B.    Assume  a  sliding  curve  eabcd,  which  may  be  a  templet  piece  & 


SLIDING   CONTACT  IN   GENERAL.  91 

cut  to  suit,  and  made  fast  to  wheel  A,  the  latter  being  also  a  tem- 
plet piece  as  shown  to  mate  with  a  templet  piece  B. 

Find  the  mating  curve  iafgh,  which,  by  sliding  action  upon 
eabcd,  will  transmit  the  same  motion,  A  to  B,  as  by  the  mutual 
rolling  of  the  rolling  curves  or  wheels  A  and  B. 

Making  Cm  —  Cj,  mg  =jc,  etc.,  we  will  have  mating  points  / 
and  m,  c  and  g,  etc.,  in  the  peripheries  of  the  non-circular  wheels. 
That  is,  these  points  will  corne  to  contact  in  pairs,  and  in  succes- 
sion, on  the  line  of  centers  as  the  non-circular  wheels  roll  in 
mutual  contact. 

Following  Fig.  106,  we  draw  a  normal  Ca  to  the  templet  G, 
when  a  becomes  one  point  in  the  required  mating  curve  for  B. 

From  j  draw  the  normal  jb  to  the  templet  G.  This  normal 
makes  a  certain  angle  with  the  radius  Aj.  Then  draw  mf,  making 
the  same  angle  with  Bm  produced,  and  make  mf  =  jb.  Then 
when  the  wheels  revolve  till  j  and  m  meet  on  the  line  of  centers, 
the  line  mf  will  coincide  with  jb,  and  the  point  /with  b.  There- 
fore,/is  a  second  point  in  the  sliding  curve  for  wheel  B.  Evidently 
g  mates  with  c  as  a  third  point.  Also  h,  i,  and  c  are  other  points 
found  in  the  same  manner  as  was  /;  oi  being  equal  to  le,  and  the 
angle  Boi  being  180°  —  Ale.  Hence  the  sliding  curve  iafgh  may 
be  drawn  in,  and  a  templet  cut  to  the  same,  to  be  mounted  upon 
the  wheel  or  templet  B.  These  sliding  curves  or  templets  thus 
made  fast  to  their  respective  wheel  templets  A  and  B  will  evidently 
work  harmoniously  and  agreeably  throughout  with  the  rolling 
curves  or  wheel  templets  A  and  B,  and  hence  with  a  common  law 
of  angular  velocity. 

The  extent  of  movement  of  these  sliding  curves  is  often  limited,, 
but  a  series  of  pairs  may  be  employed  which  engage  in  action  in 
succession,  as  in  the  case  of  the  teeth  of  gear  wheels,  where  the 
rolling  curves  tangent  at  C  answer  for  pitch  lines,  and  the  sliding 
curves  tangent  at  a  for  tooth  curves.  We  therefore  next  take  up 
the  somewhat  extensive  subject  of 

THE  TEETH   OF  GEAR  WHEELS. 

The  sliding  curves  or  tooth  curves  brought  out  above  are  some- 
times called  odontoids,  from  the  Greek  word  for  tooth.  Also  the 
rolling  curves  A  and  B  in  contact  at  C,  the  pitch  lines  in  gearing, 
are  sometimes  called  centroids,  from  the  fact  that  ICj,  etc.,  are 
points  of  instantaneous  axes  of  rolling  motion  of  the  describing- 
templet  of  Fig.  109  to  be  described. 


92  PRINCIPLES  OF  MECHANISM. 

THE  TBACING  OF  SLIDING  CURVES,  OB  ODONTOIDS,  BY  MEANS 
OF  TEMPLETS  BOLLED  UPON  NON-CIBCULAB  WHEEL  PITCH 
LINES  OB  CENTBOIDS. 

Templets  are  very  useful  aids  in  the  study  of  or  actual  drawing 
of  tooth  curves  for  gearing,  and  we  may  employ  pitch  templets, 
tooth  templets,  and  describing  templets.  Thus,  a  piece  cut  from 
thin  wood  to  the  form  AlCjck,  Fig.  108,  may  be  called  a  pitch 
templet,  and  sometimes  two  pieces  are  prepared,  one  convex  and 
the  other  concave,  fitting  each  other  along  the  pitch  line  ICk, 
either  of  which  may  be  used  at  pleasure.  Likewise,  pitch  templets 
may  be  prepared,  fitting  the  pitch  line  OCmgn.  Again,  a  piece, 
G,  cut  to  the  form  of  the  tooth  curve  ead,  and  another  for  the 
mating  tooth  curve,  may  be  called  tooth  templets.  Besides  these, 
a  describing  templet  may  be  employed  which,  in  Fig.  108,  rolled 
along  on  the  curve  or  templet  ICjc,  will,  by  an  attached  marking 
point,  trace  the  tooth  curve  eabc,  or  again,  rolled  on  oCmg,  describe 
the  mating  curve  iafg. 

Having  given  the  non-circular  wheel  pitch  curve  ICjc,  to  find 
the  describing  templet  which,  rolled  along  on  the  upper  side  of  the 
pitch  curve,  will  describe  or  generate  the  tooth 
curve  eabc:   Draw  aj,  aC,  al,  Fig.  109,  of  the 
same  lengths   as   bj,  aC,  el,   Fig.   108,    and   at 
intervening  angles  such  that  1C,  Cj,  etc.,  Fig.  109, 
shall  equal  the  like  lettered  quantities  of  Fig.  108. 
Then  draw  in  the  curve  ajCl,  Fig.  109,  and  we 
have  the  figure  of  the  describing  templet  sought, 
to  complete  which  describing  templet  cut  a  piece 
of  thin  wood  to  the  form  of  curve  found  and  set  a  marking  point 
at  a. 

To  trace  the  tooth  curve  eabc,  Fig.  108,  with  this  templet: 
Place  the  tracer,  a,  of  the  templet  at  c,  Fig.  108,  and  proceed  to 
turn  the  describing  templet  left-handed,  causing  it  to  roll  along  the 
pitch  line/C7  without  slipping,  when  the  points  j,  C,  I,  etc.,  of  the 
describing  templet,  Fig.  109,  will  fall  at  the  points/,  C,  I,  etc.,  Fig. 
108,  by  reason  of  the  equality  of  spaces,  step  by  step;  and  the  tracer 
point  a  will  at  like  points  of  time  fall  at  I,  a,  e,  etc.,  by  reason  of  the 
equality  of  lengths  of  the  lines  ja,  Ca,  la,  etc.,  Fig.  109,  with  the 
normals  jl,  Ca,  le,  etc.,  Fig.  108,  and  the  equality  of  the  angles  in 
the  two  figures  between  arcs  and  lines  at  the  points  JCI,  etc.  This 
rolling  of  the  describing  templet  may  suppose  that  a  concave  pitch 


SLIDING   CONTACT   IN   GENERAL.  95 

templet  fitting  the  pitch  line  cjCl  is  employed  for  the  describing 
templet  to  roll  upon,  and  the  curve  cae  traced  upon  a  plane  which 
is  fixed  with  reference  to  the  pitch  templet.  In  the  illustrations, 
comparatively  few  points,  c,j,  C,  I,  etc.,  are  indicated,  in  order  to 
secure  clearness  of  the  figures. 

For  the  same  reasons  that  the  tooth  curve  cbae  is  traced  by 
rolling  the  describing  templet,  Fig.  109,  on  the  pitch  line  cjCl, 
likewise  the  mating  tooth  curve  gfai  is  traced  by  rolling  the  same 
describing  templet  on  the  pitch  line  gmCo. 

It  is  to  be  observed  that  the  above  describing  templet  applies 
for  only  the  portions  of  the  tooth  curves  of  Fig.  108  which  lie 
above  or  to  the  left  of  the  pitch  lines.  For  the  portions  cd  and  gh, 
at  the  right  or  below,  a  second  describing  templet,  found  as  was 
Fig.  109,  would  generally  be  required  where,  as  in  Fig.  108,  one 
full  tooth  curve,  e  to  d,  was  assumed.  In  practice,  however,  instead 
of  assuming  the  tooth  curve,  the  describing  templet  may  advisably 
be  assumed,  and  of  any  preferred  form;  and  the  same  is  often 
applied  on  both  sides  of  each  pitch  line. 

NAMES,   TERMS,   AND    RULES   FOR   GEAR  TEETH. 

As  the  names  of  the  various  parts  of  toothed  gearing  will  be  of 
frequent  use,  they  are  here  given,  in  connection  with  Fig.  110, 


FIG.  110. 

for  future  reference.     It  will  be  a  help  also  in  the  study  of  teeth 
that  the  proportions  be  familiarized. 


'94  PRINCIPLES   OF   MECHANISM. 

A   common   rule  which   may  be   deviated  from  according  to 
the  judgment  of  the  designer  is: 

For  Circular  or  Circumferential  Pitch. 

Pitch  =  distance  from  center  to  center  of  teeth  on  pitch  line. 

Addendum  =  3/10  pitch;  working  depth  =  twice  addendum. 
Dedendum  =  4/10  pitch,  or  subaddendum. 
Thickness    =  5/11  pitch. 
Space  =  6/11  pitch. 

For  Diametral  Pitch.— For  Cut  Gears. 

A 

Pitch  =  portion  of  diameter  to  each  tooth,  =  diameter  divided 

by  number  of  teeth  in  the  wheel. 
Addendum  =  one  diametral  pitch. 

Dedendum  =  one  diametral  pitch  plus  1/7  diametral  pitch. 
Thickness    =  nearly  1.57  diametral  pitch. 


CHAPTER  IX. 


TOOTHED  GEARING  IN  GENERAL. 

DIRECTIONAL  RELATION  CONSTANT.     VELOCITY-RATIO 

VARYING. 

TOOTH  CURVES  FOR  NON-CIRCULAR  GEARING. 

CASE  I.     AXES  PARALLEL. 

ALL  correct  working  gearing  of  whatever  name,  circular  or  non- 
circular  in  outline  of  wheel,  must  possess  tooth  outlines  which  are 
describable  by  the  principles  indicated 
in  Figures  108  and  109. 

The  most  general  case  is  taken  up 
here,  instead  of  several  proceeding 
from  the  simpler  to  the  more  general, 
because  the  general  case  is  as  readily 
comprehended  as  any  special  limited 
one.  Thus,  the  whole  general  theory 
of  tooth  outlines  for  gearing  will  be 
mastered  at  once. 

Templets. 

The  theory  and  practice  for  this  case 
can  probably  be  best  explained  by  the  aid 
of  templets,  a  complete  set  of  which, 
representing  the  general  case  of  non- 
circnlar  gears,  is  shown  in  Fig.  111. 
Portions  of  the  non-circular  rolling 
curves  or  pitch  lines  are  shown  as  A 
and  B  ;  A1  being  a  templet  fitting  the 

curve   A   on  the  inside,   A9  a  second   templet  fitting  it  on  the 
-outside,  both  of  which  will  fit  together  on  the  pitch  line  A.    When 

95 


96 


PRINCIPLES   OF   MECHANISM. 


thus  together,  mark  the  point  C,  £,  and  bu  thus  made  coinci 
dent ;  also  £l  and  B^  are  like  fitting  templets  for  the  pitch  line 
B,  with  points  C,  clf  c^  marked.  These  marked  points  are  all 
mutually  mating  points  as  between  templets  or  pitch  lines.  These 
are  the  pitch-line  templets,  and,  for  brevity,  may  be  called  pitch 
templets. 

The  smaller  figures  show  the  describing  templets,  aF  and 
bG,  with  the  marking  point  at  a  and  b,  and  have  their  represen- 
tatives, in  theory,  in  Fig.  109.  All  these  templets  are  supposed 
to  be  cut  from  thin  pieces  of  wood,  metal,  or  other  suitable 
material. 


USE   OF   TEMPLETS   IN    DESCRIBING    TOOTH    CURVES. 

In  Fig.  112,  the  describing  templet  aF  is  shown  as  being  rolled 
along  the  pitch  templet  £9,  while  the  marking  point,  a,  is  describing 

the  tooth  curve  c^a  ;  also  the  same 
describing  templet  is  shown  as  being 
rolled  along  the  pitch  templet  A^  and 
describing  the  tooth  curve  bta.  Now, 
if  no  slipping  occurs  in  the  rolling,  and 
it  be  continued  on  both  pitch  lines  till 
the  arc  c^E  be  made  equal  to  Z>,Z>,  then 
will  D  and  E  be  mating  points  in  the 
non-circular  rolling  arcs  or  pitch  lines, 
and  aE  will  equal  aD  both  as  to  arc 
and  chord,  and  the  segmental  figures 
aE  and.  aD  are  equal  in  every  respect. 
In  this  rolling,  the  tooth  curves  b^a 
and  c}a  are  supposed  to  be  traced  by 
the  marker  a,  and  if  extended,  would  give  the  dotted  lines  above 
a,  a.  These  may  be  supposed  to  be  traced  upon  the  same  paper 
that  the  pitch  lines  are  drawn  upon,  and  that  the  paper  is  trans- 
parent. 

Now  if  we  place  one  drawing  upon  the  other  so  that  they  will 
be  in  common  tangency  with  the  points  E  and  D  coinciding,  then 
the  lines  aD  and  aE  will  also  coincide  with  the  two  points  a  in 
common  ;  and  the  tooth  curves  bta  and  c,«  will  touch  each  other 
at  a.  The  resulting  diagram  for  this  will  be  as  shown  in  Fig.  113; 
the  two  figures  of  the  entire  describing  curves,  indeed,  coalescing 
into  one  at  aDEF. 


FIG.  112. 


TOOTHED    GEARING    IN    GENERAL. 


FIG.  113. 


The  two  tooth  curves  b^d  and  c^ae  will  be  tangent  to  each 
other  at  a,  instead  of  intersecting;  because  it  is  evident  that  if  the 
describing  templet  aDEF  be  rolled  either 
way  on  the  pitch  lines,  the  point  a  will  move 
to  lower  positions  when  the  rolling  is  on  Ab^  B 
than  when  on  Bc^  Also,  it  is  plain  that  the 
line  aDE  is  a  common  normal  to  the  tooth  A, 
curves  b^ad  and  c^ae,  since  the  point 
serves  as  a  momentary  center  about  which  to 
describe  small  portions  of  the  curves  btd  and  c^e  at  a,  as  in  describ- 
ing a  circle  about  its  center.  These  facts  are  confirmed  by  refer- 
ence to  Fig.  108. 

We  observe  further  that  the  tooth  curves  b^ad  and  ctae  are 
wholly  above  their  respective  pitch  lines  and  the  point  of  contact,  a, 
to  the  right  of  the  line  of  centers  through  DE. 

The  principles  brought  out  above  being  perfectly  general,  they 
embrace  all  possible  forms  of  correct  working  teeth,  such  as  epicy- 
cloidal,  involute,  conjugate,  paraboloidal,  octoidal,  etc. 

Hence  all  correct  working  tooth  curves  must  possess  the  follow- 
ing properties : 

1st.  They  must  continually  remain  in  mutual  tangency. 
2d.  The  mutual  tangent  point  of  a  given  pair  of  tooth  curves 
must  remain  on  the  same  side  of  the  line  of  centers  and  of  the 
pitch  lines. 

3d.  The  straight  line  joining  the  point  of  mutual  tangency  of 
the  tooth  curves  and  of  the  pitch  lines 
must  be  a  common  normal  to  the  tooth 
curves;  and  these  normals  should  never 
intersect  between  the  tooth  curves  and 
their  pitch  lines,  and  they  must  always 
intersect  or  be  tangent  to  their  pitch 
lines. 

As  Fig.  113  provides  for  contact 
only  on  the  one  side  of  the  line  of  centers 
and  above  the  pitch  lines,  contact  below 
and  to  the  left  may  be  obtained  by 
using  the  same,  or  any  other,  desciib- 
ing  templet  below  and  to  the  left, 
as  in  Fig,  114,  where  the  other  tem- 
plet, bG,  Fig.  Ill,  is  used,  and  rolled  on 
A  and  B  below  as  shown  in  Fig.  114  and  combined  as  in  Fig, 


FIG.  114. 


98 


PRINCIPLES    OF    MECHANISM. 


115,  making   the    starting  points,   Z>a   and  ca,  coincide  with  ^c,, 
Fig.  113. 

This  work,  being  all  similar  to  that  of  Figures  112  and  113, 
will  not  require  detailed  description. 

The  Tooth  Profile. 

When  combined  with  the  points  bj>%9  c^,,  in  common  at  C, 
at  the  line  of  centers,  and  omitting  the  describing  templets,  we 
have  Fig.  116,  in  which  LCM  is  a  complete  tooth  profile  for 
one  side  of  a  tooth  of  the  wheel  A,  and  NCO  a  tooth  profile 
for  one  side  of  a  tooth  of  the  wheel  B.  Now,  as  the  wheels 
turn  one  way  from  the  position  shown,  the  tooth  contact  moves 

L   «J 


FIG.  115. 

out  in  one  direction  from  C9  or  the  opposite  direction  for  the 
opposite  turning  ;  from  which  it  appears  that  in  continuous  re- 
peated revolutions  of  the  wheels  A  and  B  there  will  be  contacts  of 
the  tooth  curves  as  they  approach  the  line  of  centers  and  also  in 
recession  therefrom. 

These  tooth  curves  in  full  lines  in  Fig.  116,  correlated  with 
other  suitable  ones  in  dotted  lines,  will  furnish  the  outlines  of  a 
pair  of  teeth  as  indicated  in  Fig.  116. 


POSITION   OF   TRACING    POINT   ON   DESCRIBING    CURVE. 

The  curves  b^ad  and  ctae,  Fig.  113,  may  be  called  trochoids 
when  the  curves  A  and  B  are  not  circular,  the  former  being  an 
epitrochoid  and  the  other  an  hypotrochoid. 

When  the  tracing  point,  a,  is  not  on  the  arc  of  FD9  but 
within  or  without,  the  curve  generated  is  called  respectively  the 
prolate  or  curtate  epitrochoid  or  hypotrochoid. 

The  tooth  curves  of  Figures  113,  115,  and  116  do  not  represent 
the  most  general  case  possible,  since  the  points  a  and  b  are  there 
carried  on  the  curve,  while  it  may  be  within  or  without.  There- 


TOOTHED   GEARING   IN   GENERAL. 


99 


fore,   let  us  examine  the  curtate  and  prolate  curves  in  quest  of 
advantages  over  those  of  Figures  113  to  116. 

1st.   The  Curtate  Trochoidal  Tooth  Curve. 

Fig.  117  represents  the  case  of  the  curtate  curves,  the  tracer, 
a,  being  carried  on  an  overreaching  piece  Ka,  fastened  to  the 
describing  curve  FK,  aK  being  normal  to  KDEF. 

Take  L  and  N  as  mating  points  in  the  pitch  curves  A  and  B, 
at  which  K  is  placed  in  starting  the  tooth  curves,  or  trochoids, 
v  Gla  and  HJa.     Then  LG  will  be  on  a 

normal  to  A,  and  UN  on  a  normal  to  B, 
and  it  is  found  that  the  portion  ///  of 
HJa  will  interfere  with  more  or  less  of 
the  curve  Gla,  showing  that  the  portion 
HJ  cannot  be  used  as  a  tooth  curve. 
With  HJ  removed,  Gl  may  be  also,  as 
FIG.  117.  useless. 

These  curtate  trochoidal  curves  from  /  and  J  upward  will 
serve  for  tooth  curves,  and  also  others  below  A  and  B,  but  they  will 
be  found  seriously  inadmissible  from  the  fact  that  there  will  be  a 
wide  space  on  either  side  of  DE  within  which  actual  working 
tooth  contacts  are  impossible,  the  very  space  where  such  contacts 
are  by  far  the  most  desirable  and  valuable.  The  limit  of  this 
space  to  the  right  of  DE  is  found  by  placing  the  tracer  point,  a, 
at  J,  and  then  noting  the  distance  DJ.  This  will  be  found  to 
increase  with  aK,  and  vanish  only  as  aK  disappears. 

2d.  The  Prolate  Trochoidal  Tooth  Curve. 

In  Fig.  118,  the  tracer,  «,  is  placed  on  the  normal  Ka,  then  with 
L  and  N  mating  points  at  which  K  starts,  as  KDEF  rolls  on  A 
and  B  respectively,  we  will  obtain  the 
proposed  tooth  curves,  Ga  and  Ha. 

These  curves  will  not  interfere  and 
may,  theoretically,  serve  for  teeth,  but, 
practically,  the  curves  are  so  nearly  par- 
allel to  the  pitch  lines  A  and  B  that  they 
will  be  more  likely  to  slip  and  wedge  the 
wheel  apart  with  serious  crowding,  than 
to  give  desirable  driving  action.  In  other 
words,  practical  tooth  curves  should  be 
perpendicular  to  the  pitch  lines  rather 
than  parallel  to  it,  and  hence  the  prolate  trochoidal  curves  of  Fig. 


100  PRINCIPLES   OF   MECHANISM. 

118  are  inadmissible,  though  less  so  the  shorter  aK  is  made  until 
it  vanishes. 

Hence  the  most  favorable  position  of  the  tracing  point  or  marker, 
a,  is  that  it  be  exactly  on  the  describing  curve  itself,  as  in  Figures 
113  and  115. 

Form  and  Size  of  Describing  Curve,  or  Templet. 

Modification  of  the  tooth  curve,  or  odontoid,  to  a  certain 
extent  may  be  effected  by  employing  one  or  another  form  of 
describing  curve. 

Thus,  the  describing  curve  may  be  a  circle,  ellipse,  parabola, 
cissoid,  spiral  of  many  turns,  or  any  arbitrarily  assumed  free-hand- 
drawn  curve,  each  giving  its  own  peculiar  tooth  curve  when  used 
on  the  pitch  lines  A  and  B  ;  the  flank  curves  apparently  varying 
more  than  the  face  curves,  as  due  to  the  changes  in  the  describing 
curve  as  illustrated  by  the  use  of  a  small  describing  curve,  la' ,  or 
large  one,  Ja",  giving  rise  to  the  flank  tooth  curves  Da'  and  Da" 
respectively. 

For  gears  engaging  externally  the  faces  will  always  be  found 
as  lying  between  the  involute  and  pitch  line,  while  the  flanks  may 
be  anywhere  from  the  pitch  line  on  the  one  side  to  the  pitch  line 
on  the  other. 

Thus,  in  Fig.  119,  aD  is  the  involute,  described  by  rolling  the 
straight  line  Fa  on  the  pitch  line  A,  the  mating  pitch  line,  B, 

being  in  this  case  straight,  as  for  a 
rack,  in  order  that  Fa  may  be  used 
to  develop  the  flank. 

The  curves  above  mentioned 
rolled  along  DF,  starting  the  tooth 
curves  at  />,  will  evidently  trace 
them  all  more  or  less  below  the  in- 
volute DG,  according  to  size,  the 
smaller -ones  when  re-entrant  mak- 
ing one  loop  after  another,  accord- 
ing to  number  of  convolutions  on  A. 
119. 

These  curves  will  all  be  normal 

to  A  at  D  except  those  developed  by  spirals  which  have  an  infinite 
number  of  coils  around  the  marking  point  a,  as  in  the  case  of  the 
logarithmic  spiral  S  with  a  at  the  pole.  Thus  the  so-called  invo- 
lute teeth,  with  involute  bases  within  A,  have  tooth  curves  which 
intersect  A  at  an  angle.  Involute  gear  teeth,  which  at  first  seem 


TOOTHED   GEARING   IN    GENERAL. 


101 


to  be  an  exception  to  the  general  theory  of  Figs.  113  and  115,  are 
seen,  nevertheless,  to  fall  within  the  general  theory,  the  tooth 
curves  being  generated  with  a  log.  spiral  describing  curve,  rolled 
outside  of  A  for  faces,  and  inside  for  flanks,  these  flank  curves 
being  terminated  exactly  at  the  base  circle,  because  here  the  radius 
of  curvature  of  the  log.  spiral  just  equals  that  of  the  pitch  circle  A, 
upon  which  it  rolls  internally. 

Example  of  Individually  Constructed  Teeth. 

In  non-circular  gears,  where  the  curvature  of  the  pitch  lines 
continually  varies,  no  two  tooth  curves  will  be  alike,  so  that  every 
one  must  be  described  individually  to  be  strictly  correct.  Fig. 
120  is  an  example  of  a  carefully  made  drawing  of  such  gears,  for 


FIG.  120. 

which  the  describing  templets  used  for  every  face  and  flank  were, 
for  convenience,  all  circles,  but  varied  in  size  according  to  the 
above  principles,  to  favor  the  shapes  of  the  teeth  to  prevent  their 
being  too  weak  across  their  flanks  at  points  of  sharpest  curvature 
of  pitch  lines. 

Non-circular  Invohite  Gears. 

In  this  case,  as  well  as  in  circular  involute  gears,  it  is  most  con- 
venient to  make  use  of  involute  bases,  or  curves,  a  little  within  the 
pitch  lines  to  which  the  tooth  curves  are  involutes,  as  above  sug- 
gested, instead  of  using  rolling  spirals. 


102  PRINCIPLES   OF   MECHANISM. 

Let  Fig.  121  represent  a  pair  of  non-circular  pitch  lines  to  be 
set  with  involute  teeth,  and  let  the 
curves  P,  I,  J  inside  of  A,  also  R,  L,  M 
inside  of  B,  be  the  bases  of  the  invo- 
lutes. The  pitch  lines  are  in  contact 
at  C,  and  for  this  position  of  wheels 
the  bases  of  the  involutes  must  be  such 
that  a  common  tangent  PR  will  pass 
through  the  point  C,  and  not  only  so  for 
this  one  position,  but  for  all  positions 
of  the  wheels  throughout  the  full  ex- 
FIG.  121.  tent  of  their  rolling,  whether  segmental 

or  otherwise. 

The  inclination  of  the  liuePCR  is  arbitrary,  except  that  undue 
irregularities  of  the  base  curves  may -be  avoided.  In  circular  gear- 
ing the  base  curves  are  usually  concentric  circles,  but  here  they 
will  usually  be  nearest  to  the  pitch  lines  at  points  of  sharpest 
curvature. 

Probably  the  best  way  is  to  place  the  pitch  lines  in  various 
positions  of  mating  tangency,  draw  a  line  PCR  for  each,  and  then 
trace  in  the  enveloping  curves  P,  7,  J  for  each  wheel.  Then  the 
involutes  SCDE  and  TCFG  may  be  drawn  for  the  teeth.  It  is 
seen  that  the  involutes  cannot  extend  within  the  base  curves  as 
at  8  and  T.  If  the  addenda  call  for  more  room,  the  space  between 
teeth  may  be  arbitrarily  deepened  below  S  or  I7  to  a  suitable  root- 
line  depth ;  but  proper  tooth  contacts  cannot  go  below  these  points, 
nor  beyond  the  tangent  points  P  and  R,  by  reason  of  interference. 
These  teeth  are  less  advantageous  for  gears  of  considerable 
eccentricity  than  those  of  epitrochoidal  form  of  Fig.  120,  because 
of  the  greater  crowding  action  explained  in  Fig.  128,  but  they  have 
the  advantage  of  requiring  less  attention  for  maintaining  exactly 
the  distance  A  B  between  centers. 


Conjugate  Teeth. 

This  practical  method  of  constructing  tooth  curves  where  one 
of  a  pair  is  arbitrarily  assumed,  and  the  other  determined  from  it. 
would  seem  to  contradict  the  refinements  of  the  above  theory  to  the 
extent  that  one  tooth  may  be  compelled  to  take  any  preferred  form 
under  sheer  caprice. 

Thus,  prepare  the  pitch  templets  A  and  B,  Fig.  122,  and  fasten 


TOOTHED    GEARING 


GENERAL. 


103 


to  B  a  sheet  of  paper  Z>,  and  to  A  a  tooth  templet  EF  elevated 
sufficiently  above  A  to  allow  the  paper  D  to  go  in  between 
A  and  F. 

Then  bring  the  two  templets  A  and  B  into  proper  contact  at  0, 
Fig.  123,  and  while  held  securely  draw  the  pencil  along  the  tooth- 


FIQ.  122. 


FIG.  123. 


curve  edge  of  the  tooth  templet  ECF,  copying  its  shape  upon  the 
paper  D.  Then  roll  A  upon  B  some  distance  without  slipping,  as 
indicated  by  the  dotted  line  at  A',  and 


B 


FIG.  124. 


while  held  fast,  draw  another  line  along 
the  edge  of  ECF.  Thus  draw  several 
lines  for  A  in  positions  to  the  right  and 
left,  when  D  will  present  a  series  of  lines, 
shown  dotted  in  Fig.  124.  Now,  the  en- 
veloping curve  GCH,  or  full  line  just 
touching  all  the  dotted  lines  on  D,  will  be  the  best  mating  tooth 
curve  for  ECF.  All  the  tooth  curves  for  one  complete  wheel 
might  be  thus  assumed,  and  the  mating  ones  for  the  other  wheel 
found  in  the  same  way. 

This  process  is  seen,  after  all,  to  present  nothing  but  what  is 
disclosed  in  Fig.  108,  since  G  there  corresponds  with  .Fin  Fig.  123, 
so  that  a  similar  process  carried  out  in  Fig.  108  would  result  in  the 
curve  hgfai. 

Limited  Inclination  of  Tooth  Curves. 

In  the  practice  of  designing  non-circular  gears  of  considerable 
eccentricity,  it  will  be  found  very  desirable  at  times  to  make  the 
teeth  nearly  radial  and  oblique  to  the  pitch  curves,  as  in  the  full 
lines  of  Fig.  125,  instead  of  normal  to  the  pitch  curves  and  in- 
clined to  the  radius;  thus  to  prevent  the  tendency  of  the  teeth  to 


104 


PRINCIPLES   OF    MECHANISM. 


slip  out  of  engagement,  as  well  as  the  excessive  crowding  apart  of 
the  gears. 

Therefore,  let  Fig.  125  represent  portions  of  a  pair  of  rather 
eccentric  non-circular  pitch  curves  A  and  B,  with  a  proposed  pair 
of  teeth  E  and  G  in  contact  at  the  line  of  centers  C,  which,  in  the 


FIG.  125. 

light  of  assumed  tooth  curves  of  Fig.  122,  and  in  spite  of  the  theory 
previously  detailed,  are  set  nearly  radial  to  avoid  crowding. 

Now,  with  B  stationary,  roll  A  around,  without  slipping  on  B 
to  the  dotted  position  A',  with  E  following  around  to  Ef,  shifting 
the  contact  of  the  pitch  curves  from  C  to  D.  Then  it  will  be  found 
that  the  tooth  E  at  E'  fatally  interferes  with  the  tooth  G,  showing 
at  once  that  the  amount  of  departure  from  the  pitch  curve  normal 
toward  the  wheel  radii  shown  in  the  figure  is  absolutely  prohibited. 
In  some  actual  wheel  designs  where  this  partiality  for  the  radius 
existed,  the  teeth  necessarily  were  doctored  before  the  wheels 
would  work. 

Thus,  in  Fig.  126,  we  have  a  photo-process  copy  of  a  pair  of 
gears  made  for  practical  use  in  machines,  examples  of  which  were 
exhibited  at  the  Centennial  of  '76,  in  which  an 
effort  at  inclining  the  tooth  toward  the  radius, 
giving  it  a  hooklike  form,  was  evidently  attempted, 
but  was  vetoed  by  the  theory  when  the  wheels  were 
put  to  work,  as  is  plain  by  the  fact  that  the  hook 
feature  was  removed  from  the  hook  or  overhanging 
side.  One  interesting  fact  incidentally  connected 
with  these  wheels  is  that  they  are  cut  gears,  the 
gear  cutter  to  cut  them  having  twenty-five  cutters. 
Thus  we  have  a  rare  example  of  other  cut  non- 
circular  gears  than  elliptic. 

Other  examples  of  the  endeavor  to  obtain  hook- 
like  teeth  were  found  at  the  Centennial;  one  show- 


FIG.  126. 


TOOTHED    GEARING 


GENERAL. 


105 


ing  this  peculiarity  in  a  moderate  degree  being  illustrated  in  Fig. 
127. 

Another  example  was  found  in  a 
shingle-sawing  machine  from  Wisconsin. 

In  all  the  above  discussions,  from  Fig. 
92  on,  we  have  treated  the  case  as  if  the 
wheels  were  plane  figures  without  thick- 
ness, while  in  reality  we  may  suppose  them 
to  have  any  arbitrary  thickness  as  cylin- 
ders, transverse  sections  of  which  form  the  above  figures. 


FIG.  127. 


Nearest  Approach  of  Tooth  Curve  toward  the  Radius. 

To  determine  absolutely  the  nearest  approach  of  the  tooth  curve 
to  the  radius,  we  have  only  to  refer  to  the  principles  laid  down  as 
following  Fig.  113,  and  especially  that  of  the  intersection  of  the 
pitch  line  by  all  the  normals  to  the  proposed  tooth  curve.  Thus 
the  curve  cj),  Fig.  114,  will  become  more  and  more  erect,  with* 
respect  to  the  pitch  line  BE',  the  farther  the  intersection  E'  is 
kept  away  from  the  point  c2:  and  this  is  favored  by  a  large  curve 
bG,  the  largest  being  the  straight  line.  But  this  same  curve  must 
roll  inside  of  A,  and  hence  the  curve  A  itself  is  the  largest  possible 
one.  For  this,  however,  the  flank  Z>2&  would  be  merely  a  point,  and 
hence  objectionable  from  considerations  of  wear. 

It  therefore  appears  that  a  sharper  curve  than  A  is  desirable. 
In  Fig.  128,  A  and  B  is  a  pair  of  non-circular  wheels  where  the 

attempt  is  to  be  made  to  carry 
the  tooth  curve  GC  as  far  as  prac- 
ticable over  toward  the  direction  of 
the  radius^  exchanging  GC  to  some 
position  JVC' as  dotted. 

First,  drawing  a  normal  NO  to 
the  proposed  dotted  curve  CN9  this 
normal  is  found  to  escape  the  pitch 
curve  CL  altogether,  and  hence  the 
curve  CNis  steeper  than  admissible, 
since  the  normal  must  intersect  the 
pitch  curve  CL. 

Next,  we  propose  a  describing 
curve,  E,  of  large  radius,  such  as 
will  but  just  go  within  the  mating 
pitch  curve  CM  and  roll,  giving  the  shortest  admissible  flank  Cl. 


FIG.  128. 


106  PRINCIPLES   OF   MECHANISM. 

This  describing  curve  E  must  be  accepted  as  that  which  will  give 
the  greatest  permissible  deviation  of  GC  to  ward  NO. 

This  tooth  curve  GC  is,  however,  far  from  coinciding  with  the 
direction  of  the  radius  BC,  and  hence  as  the  flank  CI  drives  against 
the  face  GC,  there  will  be  a  more  or  less  serious  crowding  apart  of 
the  wheels  A  and  B. 

Wear  in  practical  use  of  wheels  is  the  criterion  by  which  67  is 
limited  in  length,  it  being,  at  best,  much  shorter  than  CG  against 
which  it  works.  Good  judgment  would  protest  against  CI  being 
reduced  to  a  point,  as  above  suggested,  for  the  case  where  E  con- 
forms with  CM-  and  also  the  figure  shows  that  this  only  very 
slightly  moves  the  curve  CG  from  its  present  position  toward  ON. 

Hence,  though  the  utmost  possible  careening  of  CG  toward  CN~ 
occurs  when  67  is  reduced  to  a  point,  yet  practical  considerations 
favor  a  slightly  less  elevated  curve  CG,  to  prevent  an  appreciable 
length  of  flank  67 ;  and  we  conclude  that  a  tooth  curve  CG,  lying 
near  the  radius  BC  prolonged,  is  impossible. 

On  the  other  hand,  the  tooth  curve  may  be  inclined  the  opposite 
way,  or  toward  the  pitch  line,  to  any  desired  extent,  as  shown  in  Fig. 
128  at  DH&ud.  DF\  it  being  simply  necessary  to  make  the  describing- 
curve,  for  example  FK=  JP,  quite  sharp  in  curvature,  or  smallr 
and  a  spiral  of  several  turns  about  the  polo  J  for  inclination  of 
tooth-curve  at  D. 

Practical  Limit  in  Eccentricity  for  Non-circular  Gears. 

In  theory  there  is  no  limit  short  of  a  radius  for  each  wheel;  but 
in  practice  friction  interposes  a  limit,  the  full  investigation  of  which 
is  out  of  place  here,  so  that  a  few  examples,  only,  will  be  cited  to- 
indicate  how  essential  are  the  principles  of  Fig.  128  in  gear 
designing. 

Fig.  129  shows  a  pair  of  gears  made  for  use  on  a  machine  to  run 
at  about  350  revolutions  per  minute.  They  are  4£  inches  between 
centers  and  have  involute  teeth,  both  being  cast  in  iron  from  one 
pattern  and  to  work  without  lubrication.  They  were  laid  out  after 
the  principles  of  Fig.  79. 

When  started  in  the  newly  developed  machine,  whole  segments 
of  the  rims  of  these  gears  would  drop  out  upon  the  floor  in  less 
than  two  minutes,  though  believed,  as  far  as  ordinary  theoretic 
strains  were  concerned,  to  be  ample  in  strength.  On  examination, 
it  was  found  that  the  involute  tooth  curves  were  too  much  inclined 
to  the  pitch  lines,  so  as  to  actually  block  the  wheels  at  positions 


TOOTHED   GEARING   IN   GENERAL. 


107 


where  the  radius  was  also  much  inclined  to  a  perpendicular  to  the 


rim. 


FIG.  129. 

It  was  found  that  elliptic  wheels  would  give  a  more  favorable 

obliquity  of  pitch  line  to  radii,  and  that  teeth  generated  by  large 

describing  curves  like  E,  Fig.  128,  would  give  easier  action;  where- 

^  upon    new   wheels   were    adopted 

^^^Jj)         ^^^^^f       w^k  these  changes  and  with  coarser 

gw;j         H^P  6^    teeth,  a  photo-process   copy   of  a 

ifr^jfl      IUI  ^^fc  Pa*r  °^  wnicn  is  given  in  Fig.  130. 

^IH  In  these  wheels  every  effort  was 

^^Bri^^^9       made  to  provide  against  breakage 

^W^  by  "blocking,"  two  considerations 

FIG.  130.  being  included  besides  those  above 

mentioned,  consisting,  first,  in  thicker  teeth  on  the  sides  of  the 

wheels  than  at  the  ends,  and,  second,  shorter  teeth,  compared  with 

pitch,  than  for  usual  proportions.     The  teeth  are  also  much  coarser 

than  in   the  previous  wheels  of   Fig.    128,  so  much  so  that   but 

little  room  was  left  for  arms  and  openings. 

A  quite  notable  case  of  very  eccentric  gears  is  given  in  Fig. 
131,  made  in  cast  iron,  2  feet  between  centers,  the  larger  being 
some  3£  feet  long,  and  used  on  a  hay-press,  where  the  larger  wheel 
is  driver.  Here  the  teeth  are  of  unequal  size  as  well  as  in  Fig. 
130,  the  proximity  of  the  axis  in  the  small  wheel  controlling  the 
size  at  the  end.  The  tendency  to  block  in  action  is  apparently 
great,  as  due  to  the  unusual  length  of  the  teeth,  they  being  here 
extended  to  sharp  points. 

An  examination  of  the  figure  indicates  that  for  the  large  wheel 


108  PRINCIPLES   OF   MECHANISM. 

as  driver  the  blocking  tendency  at  places  is  very  great,  so  much  so 
that  thorough  lubrication  must  be  a  necessity  to  bring  the  coefficient 
of  friction  down  to  permit  the  wheels  to  work  at  all. 


FIG.  131. 

Fortunately  for  this  case  the  wheels  are  used  in  a  horse-power 
machine,  where,  if  the  wheels  block  from  deficient  lubrication,  the 
horses  may  stop  as  a  sign  for  lubricant  wanted,  the  wheel  having 
sufficient  strength  to  resist  the  horses.  In  a  machine  with  the 
inertia  of  a  heavy  fly-wheel  or  other  parts  involved,  the  result  would 
be  different. 

The  full  line  drawn  as  a  common  normal  to  a  pair  of  teeth  just 
taking  ^contact,  with  the  large  wheel  driving  right-handed,  strikes 
yery  near  the  center  of  the  driven  wheel,  making  a  case  of  positive 
block  with  coefficient  of  friction  at  0.083.  As  the  coefficient  of 
friction  is  about  0.12  to  0.16  for  dry  cast  iron,  and  0.05  to  0.08  for 
lubricated,  it  is  clear  that  these  gears  cannot  run  except  with  teeth 
well  lubricated. 

In  the  case  of  the  dotted  line  for  the  next  tooth  there  is  still 
greater  doubt. 

If  these  teeth  were  cut  off  to  the  ordinary  addendum  of  3/10 
pitch,  the  tendency  to  block  would  be  very  much  less,  and  still  bet- 
ter if  cut  to  the  minimum  where  one  pair  of  teeth  engage  in  action 
as  the  preceding  pair  quits  engagement,  and  the  wheels  would  then 
probably  run  without  lubrication. 

"Whence  it  appears  that  unduly  long  teeth,  such,  for  example,  as 
run  to  a  point,  are  especially  objectionable  in  very  eccentric  non- 
circular  gears. 


TOOTHED    GEARING   IN    GENERAL.  109 

Again,  where  one  of  a  pair  is  always  to  be  driver,  the  addendum 
of  the  driven  wheel  may  with  advantage  be  made  unduly  short  with 
that  of  the  driving  wheel  increased. 

In  heavy  gearing  rigidly  connected  with  heavy  revolving  parts, 
the  ratio  AQ  over  QH,  Fig.  128,  where  QH  is  a  normal  to  (7//at 
H9  should  not  exceed  0.4  to  0.5  for  teeth  running  without  lubrica- 
tion. 

It  therefore  appears  that  there  is  a  practical  limit  of  eccentricity 
in  non-circular  gearing,  occasioned  by  the  "blocking"  tendency  at 
points  where  the  pitch  line  departs  farthest  from  a  normal  to  the 
radius,  and  that  this  limit  is  favored  by  short  addendum  of  teeth 
of  driver;  by  large  tooth-generating  curve  for  side  of  tooth  in  ques- 
tion; by  thick  and  stout  teeth;  by  lubrication;  and  by  giving  the 
driver  a  slightly  greater  pitch  than  follower. 

Substitution  of  Pitch-line  Rolling  for  Teeth  in  Extremely 
Eccentric  Gears. 

In  some  cases  of  very  eccentric  gears,  where  blocking  would 
surely  occur  with  teeth,  the  latter  may  be  omitted  with  advantage 
and  the  driving  action  from  driver  to  fol- 
lower effected  by  the  mutual  rolling  of  the 
pitch  lines. 

Fig.  132  is  an  example  of  this,  which 
represents  the  limiting  case  of  the  "quick 
return  "  movement,  where  a  crank  and  pit- 
man is  made  to  drive  the  connected  slide 
forward  at  a  uniform  velocity  and  back  in 
much  less  time,  or  "quick."  The  model 
of  Fig.  132  represents  the  theoretical  limit 
of  uniform  motion  forward  from  beginning 
to  end,  and  back  in  an  absolute  instant,  or 
no  time  at  all.  This  is  evidently  impos-  plG  i^ 

sible,  since  the  driven  wheel  must  make  a  half-turn  in  the  absolute 
instant,  during  which  the  driver  would  be  stationary  with  no  power 
to  drive. 

In  the  model  this  difficulty  is  removed  by  placing  the  handle 
on  the  driven- wheel  crank  or  lower  wheel  in  the  model. 

In  the  absence  of  teeth,  the  eccentric  pitch  surface  of  the 
driver  acting  upon  that  of  the  follower  can  drive  in  one  direction, 
but  not  in  the  other,  and  to  prevent  derangement  some  device 
must  be  supplied. 


110 


PRINCIPLES   OF    MECHANISM. 


In  elliptic  gears  a  link  may  be  added  as  in  Fig.  133,  wher«e 
the  eccentricity  is  too  great  for  good  working  teeth   throughout, 
so  that  a   pair,  only,  of  teeth  is  added  at  the  ends  and  the  link 
.-—7~s*f  introduced  to  hold  the  elliptic  pitch  lines  in 

^  .  rolling  contact. 

But  in  Fig.  132  a  link  cannot  be  employed, 
tUBI^^St    so  that  a  circular  sector  with  non-circular  arcs 
at  the  ends  is  added  to  engage  with  a  mating 
piece  on  the  opposite  wheel.       These  are  so 
shaped  as  to  hold  the  upper  wheel  stationary 
for  the  allotted  time  of  the  half-turn  of  the 
lower  wheel,    after    which    the   non-circular 
curves  keep  the  rolling  pitch-line  arcs  in  con 
^g5  tact  for  the  space  where  teeth  are  omitted. 

(.  Fig.  134  is  a  carefully  made  drawing  of  a 

pair  of   gears  used  for  a  while  in  a  certain 
FIG.  133.  screw  machine  for  a  "  quick  return"  to  dirve 

a  slick  uniformly  forward  and  back  with  the  least  allowable  time 
greater  than  an  absolute  instant.  It  is  produced,  in  effect,  by 
modifying  Fig.  132,  as  by  splitting  the  long  projecting  point  of  the 
upper  wheel  and  spreading  it  to  DAE  by  a  circular  sector,  mated 
with  FCG  in  the  other  wheel,  the  two  wheels  being  otherwise  modi- 
fied in  a  manner  to  preserve  the  quick-return  principle  exactly. 
As  adopted,  the  advance  occupies  3/4,  and  the  return  1/4  of  the 
time.  The  line  of  centers  is  about  five  inches. 

From  H  to  F,  and  G  to  /,  the  pitch  lines  roll  upon  the  mating 


FIG.  134. 


pitch  lines  DJ  and  EK9  respectively;   teeth  engaging  for  the  re- 
maining arcs. 

To  prevent  the  gears  from  getting  deranged  in  this  case,  pro- 
jections from  .4,  shown  by  dotted  lines  at  N  and  0,  are  added, 


TOOTHED   GEARING 


GENERAL. 


Ill 


which  fall  inside  of  projections  L  and  Mt  respectively,  and  serve 
to  keep  the  pitch  lines  approximately  in  contact,  and  assuring  the 
engaging  and  disengaging  of  teeth  for  several  hundred  revolutions 
per  minute  in  either  direction. 


FIG.  135. 


FIG.  136. 


Fig.  1 35  gives  a  better  view  of  the  projections  L,  M,  Nf  and  0 
of  Fig.  134. 

These  wheels  served  with  entire  satisfaction,  as  operating  gears, 
for  a  considerable  time  in  the  machine  they  were  designed  for. 
But  as  the  return,  though  decidedly  more  moderate  than  in  Fig. 
132,  was  yet  decidedly  too  quick  for  the  purpose,  other  wheels  with 
still  less  rapid  return  were  found  advisable,  and  obtained  in  the 
wheels  shown  in  Fig.  136. 

Here  the  return  was  so  extended  that  one  of  the  rolling  arcs 
was  dropped,  the  other  being  necessarily  re- 
tained to  preserve  the  law  of  uniform  motion 
of  the  slide,  the  one  set  of  projections  L  and  N 
being  also  retained  to  provide  against  mis- 
€ngagement  of  teeth. 

These  wheels,  as  in  Fig.  136,  are  doing  good 
service  in  extended  use  and  are  thoroughly 
practical  gears. 

Internal  Non-circular  Gears. 

These  are  practical  for  complete  wheels 
where  the  "  internal "  gear  or  that  having  teeth 
within  the  rim  is  at  least  twice  as  large  as  the 
pinion,  or  smaller  wheel;  and  the  tracing  of  the 
teeth  is  as  before  except  for  the  fact  that  FlG-  137< 

generally  both  pitch  lines  are  convex  in  the  same  direction  as  in 
Pig.  137,  so  that  a  further  description  is  unnecessary. 


CHAPTER  X. 
TEETH  OF  BEVEL  AND  SKEW  BEVEL  NON-CIRCULAR  WHEELS, 

CASE  II.    THE  GENEEAL  CASE  OF  AXES  INTERSECTING. 

WE  here  assume  that  the  pitch  lines  are  already  made  out  act- 
ually on  the  normal  spheres,  or  else  as  ordinates,  angles,  etc.,  by 
which  the  curves  may  be  drawn,  as  explained  under  Case  II  of 
Rolling  Contact,  with  Axes  Meeting. 

Let  Fig.  138.  represent  a  pair  of  such  rolling  wheels  with  cen- 
ters at  A  and  B9  and  with  C  the  common  tangent  point.  Such 


FIG.  138. 

wheels  with  axes  meeting  are  conical  as  explained  under  rolling- 
contact,  and  may  extend  in  thickness  to  the  vertex  0  of  the  cones. 
The  front  surfaces  of  the  wheel  at  AC  and  CB  are  spherical  sur- 
faces, 0  being  the  center  of  the  sphere  and  the  common  vertex  of 
the  cones.  The  finished  wheels  are  usually  as  if  cut  from  spherical 
shells  as  in  Fig.  83. 

The  describing  curve  or  templet  to  carry  the  tracing  point  is 
here  a  cone,  also,  with  base  aCF,  and  with  vertex  at  0  in  common 
with  the  other  cones.  The  tracer  or  marker  is  here  a  line  aO. 

112 


TEETH  OF  BEVEL  AND  SKEW  BEVEL  NON-CIRCULAR  WHEELS. 

Let  L  and  N  represent  mating  points  on  the  rolling  cone-wheel 
bases,  LC and  JVC"  being  equal.  Then  placing  the  line  aO  of  the 
describing  cone  coincident  with  the  element  LO  of  the  wheel 
cone,  and  rolling  it  around  to  the  position  aCF,  we  describe  a  sur- 
face LaOL,  which  may  be  extended  to  c  as  shown.  Also  placing 
the  same  describing  cone  inside  the  wheel  B,  and  with  the  line  aO 
coincident  with  the  element  NO  of  the  wheel  B  and  rolling  it 
around  to  the  position  aCF,  we  generate  the  surface  NaON,  which 
may  also  be  extended  by  further  rolling.  These  two  generated 
surfaces  will  be  tangent  to  each  other  along  the  element  aO,  and 
hence  these  surfaces  form  proper  tooth  surfaces  for  teeth  of  these 
wheels,  whatever  their  thickness. 

In  the  case  of  spherical  shells,  the  tooth  curves  may  be  traced 
directly  on  them,  both  sides  if  desired,  by  means  of  rolling  templets 
of  wood  cut  concave  to  the  spherical  form  and  of  any  preferred 
outline  a  OF. 

In  Fig.  138  we  have  a  face  for  a  tooth  of  A,  and  a  flank  for  a 
tooth  of  B,  contacts  between  which  will  always  be  inside  of  B  and 
to  the  right  of  ACB.  Contacts  on  the  other  side  may  be  obtained 
by  rolling  the  templet  aCF,  or  any  other  one,  inside  of  A  and  out- 
side of  By  thus  procuring  a  flank  for^i  and  a  face  for  B. 

For  non-circular  wheels  with  large  teeth  it  will  be  necessary  for 
accuracy  to  trace  every  individual  tooth  curve  for  the  entire  wheel, 
no  two  curves  being  alike;  but  the  same  templet  will  not  be  re- 
quired except  for  the  pair  of  tooth  curves  that  are  to  work  to- 
gether, as  La  and  Na  in  Fig.  138. 

In  small  teeth  less  care  is  needed  and  approximate  tooth  lines 
will  serve,  as,  for  instance,  for  1  or  1£  inches  pitch  or  less.  In 
some  cases,  as  for  instance  in  Fig.  98,  where  the  gears  were  cast, 
the  patterns'  had  the  rim  formed  to  the  root  line,  when  teeth 
blocked  out  for  the  full  depth  in  separate  pieces,  by  a  drawing 
giving  some  excess  of  curvature  at  the  sides,  were  fastened  upon  the, 
rim  and  the  patterns  finished  and  the  gears  cast. 

The  Tredgold's  approximate  method,  hereafter  described  fully- 
in  connection  with  circular  bevel  gears,  may  be  employed  here  with , 
sufficient  accuracy  for  any  practical  purpose. 

Involute  teeth  may  be  laid  out  by  first  finding  the  base  lines  and 
using  rolling  planes  on  the  base  lines,  similarly  as  lines  are  used  in 
Fig.  121,  the  tooth  surfaces  being  generated  by  a  line  on  the  rolling; 
plane  running  to  0  of  Fig.  1 38. 


114 


PRINCIPLES   OF   MECHANISM. 


CASE  III.    THE  GENERAL  CASE  otf  AXES  CROSSING  WITHOUT 
INTERSECTING. 

NON-CIRCULAR    SKEW  BEVELS. 

A  method  for  the  exact  construction  of  teeth  for  this  case  is 
not  known,  nor  even  for  circular  skew  bevels  for  serviceable  gears, 
except  such  as  are  taken  from  near  the  shortest  common  perpen- 
dicular between  the  axes. 

Suppose  the  pitch  curves  have  been  determined  upon  the  spher- 
ical blank  as  at  p,  Fig.  139,  by  Figures  93  to  95.  Then  the  adden- 
dum and  dedendum  lines  may  be  laid  out  as  at  d  and  e,  Fig.  139. 
Between  the  last-named  lines  the  tooth  curves  are  to  be  drawn  as 
shown  by  the  method  of  Fig.  138,  rather  small  describing  curves 
being  used  to  prevent  the  teeth  from  interfering  unduly  by  reason 
of  the  admitted  approximate  form  of  these  teeth  for  non-circular 
skew  bevel. 

Then,  as  explained  in  connection  with  Fig.  95,  construct  the 
I  second  spherical  blank  M  to  mount 

upon  an  axial  rod  equidistant  with  it 
from  the  gorge,  the  teeth  being  laid 
out  upon  M  in  the  inverse  order  with 
respect  to  A\  or  in  like  order  as  the 
blanks  are  viewed  in  Fig.  139,  the 
convex  side  of  A  and  the  concave 
side  of  M  being  towards  the  eve. 

Cut  the  teeth  on  M  to  line  and 
bevel  back  towards  the  concave  side 
so  that  the  line  or  string  CO,  drawn 
tight,  will  spring  from  the  line  of  the 
tooth  curve  on  the  convex  surface  of 
M. 

Then  the  blank  A  may  be  cut  to 
the  addendum  line  and  to  the  tooth 
curves  by  stretching  the  string 
across,  at  various  corresponding 
points  of  A  and  M,  as  at  A,  m,  n,  o,  etc.,  and  at  as  many  inter- 
mediate pairs  of  points  as  desired. 

These  teeth  will  be  twisted,  but  the  face  and  flank  surfaces  will 
be  composed  of  straight-lined  elements. 

Instead  of  cutting  these  teeth  from  the  solid  material  of  the 
blank  A,  the  latter  may  be  dressed  off  carefully  to  the  dedendum 


FIG.  139. 


TEETH  OF  BEVEL  AND  SKEW  BEVEL  NON-CIRCULAR  WHEELS.    115 

line  and  separate  blocks  for  teeth  may  be  prepared  and  fastened 
upon  the  blank,  and  finished  subsequently  unless  quite  small  and 
short  along  CD. 

The  wheel  B,  mating  with  A,  maybe  made  in  a  similar  manner. 
If  they  are  of  the  same  size  and  alike,  one  pattern  for  easting  will 
serve  for  both  A  and  B,  but  otherwise  not.  The  wheels  may,  either 
•of  them,  be  made  1-lobed,  2-lobed,  etc. 

When  a  pair  of  wheels  are  thus  far  completed  they  will  not  run 
together  smoothly  because  of  the  already-mentioned  incorrectness 
of  the  tooth  outlines  for  this  case. 

The  running  conditions  may  be  improved  by  mounting  the 
wheels  as  mated  in  mesh,  and  by  revolving  slowly  under  careful 
scrutiny,  the  high  or  interfering  points  noted,  and  dressed  off  by 
hand,  this  process  being  continued  till  the  desired  degree  of  smooth- 
ness of  running  is  obtained. 

The  resulting  wheels,  obtained  as  above,  may  be  of  wood  and 
used  for  patterns  for  casting  in  metal  with  no  greater  further 
expense  than  for  circular  wheels. 

For  wheels  of  one  inch  pitch  or  less  it  is  probable  that  teeth 
may  be  formed  in  separate  pieces  and  made  fast  upon  the  blanks  A 
and  B,  and  dressed  to  the  dedendum  line  and  surface. 


CHAPTER  XL 
INTERMITTENT  MOTIONS. 

TEETH;  TOGETHER  WITH  ENGAGING   AND   DISENGAGING 
SPURS  AND   SEGMENTS   IN   GENERAL. 

TEETH. 

WE  now  come  to  consider  the  teeth  and  spurs  of  intermittent 
motions  whether  having  circular  or  non-circular  pitch  lines. 

The  teeth  present  no  new  problems  over  segmental  or  partial 
gears,  as,  for  instance,  in  Fig.  96,  the  pitch  line  FED  when 
equipped  with  teeth  is  as  a  segmental  gear,  to  mesh  or  mate  with 
the  segmental  gear  GMH,  and  any  preferred  form  of  tooth  may  be 
adopted  and  laid  out  as  already  explained.  There  should,  however, 
be  a  space  to  start  with  at  F  and  D  and  a  tooth  at  G  and  H,  as  here 
shown  in  Fig.  140  at  N  and  K. 

Directional  Relation  Constant. 

THE   ENGAGING    AND    DISENGAGING   SPURS. 

The  spurs  GF  and  HI  are  to  start  the  driven  member  B  from 
the  full  stop  on  the  locking  arc  into  full  motion,  as  for  the  initial 
engagement  of  the  teeth,  and  this  usually  by  contact  action  of  the 
spurs.  Where  a  sufficiently  large  arc  of  movement  can  be  allowed 
to  A  in  which  B  is  to  be  started  or  stopped,  the  spurs  may  work  by 
rolling  contact  instead  of  sliding. 

In  Fig.  140  we  have  a  correct  drawing  of  a  pair  of  non-circular 
plane  wheels  of  the  intermittent  order,  showing  two  complete  sets 
of  prongs  or  spurs  for  engaging  and  disengaging.  One  set,  num- 
bered 1,  works  by  sliding  contact  both  ways,  and  the  second  set, 
numbered  2,  works  by  rolling  contact;  the  same  arc  of  movement 
ab  for  wheel  A  being  required  to  swing  B  from  rest  into  full  en- 
gagement when  the  sliding-contact  spurs  are  employed  as  when 
the  rolling  contact  spurs  2  are  in  use.  The  same  is  true  for  the 
spurs  on  either  side  of  the  wheel,  the  arc  cd  being  greater  than  ab, 
as  required  by  the  greater  drop  MN  in  radial  distance. 

116 


NON-CIRCULAR    INTERMITTENT 


The  spur  HI  mates  with  FG,  and  they  are  so  made  that  when 
the  latter  approaches  the  former,  F  and  H  are  the  first  points  to 
touch;  this  same  contact  starting  B  towards  engagement  of  the 
gear  teeth,  when  further  movement  causes  the  point  of  contact  to 
move  along  toward  G  and  1,  arriving  at  which  the  full  engagement 
of  the  gear  teeth  should  be  assured. 


\b 


FIG.  140. 

Where  this  movement  takes  place  the  circular  locking  arc  JM 
must  be  cut  away  as  at  JK,  and  to  just  such  extent  that  the  wheel 
R  shall  not  have  undue  backlash,  it  being  prevented  by  the  bearing 
of  B  upon  JK  for  the  one  direction,  and  by  the  contact  between 
FG  and  HI  for  the  other  direction.  The  same,  of  course,  is  true 
of  the  other  side  of  the  wheel  A,  where  B'  is  shown  as  bearing 
against  MN  for  the  one  direction  of  motion  of  B'  and  against  the 
spurs  II  for  the  other  direction.  It  is  essential  for  the  satisfactory 
performance  of  these  wheels  that  these  curves  and  spurs  be  well 
mated  with  respect  to  reduced  backlash. 

For  rapid  rotation  the  spurs  need  to  be  much  the  longest.  For 
slow  motion  a  pin  may  do  for  FG,  with  HI  cut  shorter  to  match,  as 
shown  at  D  and  F,  Fig.  141,  where  the  shock  is  some  4  to  6  times  as 
great  as  in  Fig.  140.  But  the  shock  is  materially  reduced  by  giving 


118  PRINCIPLES  OF  MECHANISM. 

the  point  of  the  spur  a  curve  as  shown  in  the  example  at  E  and  7/y 
Fig.  141,  the  intensity  of  the  shock  being  judged  of  by  the  perpen- 
dicular distance  from  A  to  the  normals  shown.  Some  shock  will 
occur  at  all  events  when  the  points  //  and  F,  Fig.  140,  meet  in 
contact  unless  ///and  FO  are  so  much  extended  that  the  normal 
FD  to  GF  meets  the  center  A.  To  determine  the  amount  of 
shock  in  a  given  case,  we  are  to  consider  the  segments  AC  and  bC 
of  the  lino  of  centers  Ab  at  the  initial  contact  of  //  upon  F,  these 
segments  being  formed  by  the  common  normal  FD. 

In  EP  and  QK,  Fig.  140,  we  have  a  pair  of  spurs  working  by 
rolling  contact  instead  of  sliding.  These  differ  from  the  sliding 
spurs  in  being  much  longer  for  a  like  tendency  to  shock,  and  con- 
sequently objectionable,  though  the  rolling  action  is  in  their  favor 
in  respect  to  wear.  To  determine  the  lengths  for  equal  shock  wo 
note  that  the  distance  AE  for  the  rolling  spurs  should  be  the  same 
as  the  length  A  C  for  the  sliding  spurs.  The  same  is  true  for  the 
terminals  of  the  other  spurs  at  SX  and  VT. 

The  rolling  spur  (>72,  which  mates  with  EPK,  is  unavoidably 
concave  because  the  contacts  which  occur  on  QR  must  be  progress- 
ive from  Q  toward  /?,  and  must  always  be  on  the  line  of  centers  for 
rolling  action;  and  for  a  gradual  start  of  B  from  rest,  the  line  QR 
near  Q  must  be  nearly  radial  and  recline  more  and  more  with  re- 
spect to  a  radius  as  shown  by  the  line  QR,  and,  in  fact,  should  run 
into  the  pitch  line  of  B  as  a  continuous  rolling  line.  Likewise 
EPK  should  run  into  the  pitch  line  KN. 

Again,  that  the  acceleration  be  normal,  the  curve  EP  should 
gradually  turn  over  into  the  pitch  line  for  the  teeth  at  /T,  as  if  it 
was  the  termination  of  that  line  toward  A,  both  portions  EP  and 
KN  being,  in  fact,  a  continuous  rolling  arc,  and,  reasonably  enough, 
should  begin  at  E  near  A,  and  by  continuous  smooth  curvature 
pass  the  points  P,  A",  and  on  to  N.  Thus  this  half  of  these  wheels 
resembles  the  half  of  the  wheels  of  Figs.  132  and  134,  where  a  pair 
of  rolling  curves  run  into  the  pitch  lines  of  gear  teeth. 

At  MN  the  locking  arc  is  necessarily  flexed  to  a  greater  dis- 
tance toward  A,  by  reason  of  the  non-circular  pitch  line  NK  em- 
ployed, and  MN\&  shaped  as  due  to  an  effort  to  give  B  a  quick 
move  from  lock  to  engagement,  as  may  sometimes  be  necessary  in 
the  application  made  of  these  wheels.  To  this  end,  it  is  here  found 
necessary  to  give  B  a  more  rapid  motion  for  a  part  of  this  turn 
from  rest  to  engagement,  than  it  will  have  while  engaged  with  the 
teeth.  This  is  not  so  evident  from  the  sliding  spurs  8  and  JTas 


NON-CIRCULAR  INTERMITTENT   MOTIONS.  1,19 

from  the  rolling  ones  777JVand  VW,  for  which  the  law  of  angular 
velocity  is  made  the  same.  Examining  the  latter,  we  note  that  for 
contact  of  VWon  TU  from  T  up  to  U  the  acceleration  is  rapid, 
and  that  at  U  the  speed  of  B  is  full  double  that  due  to  contact  at 
j^when  the  gear  teeth  have  fairly  engaged.  One  effect  of  this  is 
to  leave  the  tooth  Y  behind  with  liability  of  its  interfering  with  the 
tooth  it  is  approaching.  Again,  it  appears  prejudicial  to  accelerate 
the  motion  of  B  to  100  per  cent,  in  excess  and  retard  it  again,  the 
only  excuse  for  which  in  practice  is  probably  to  be  found  in  a 
necessity  for  a  quick  movement  of  B  from  rest  to  the  full  engage- 
ment of  the  gear  teeth,  or  for  a  comparatively  small  arc  cd  for  it. 

At  all  events  it  is  certain  that  the  rolling  spurs  TYAVand  VW 
are  utterly  impracticable  at  least  from  U  to  N,  where  the  driving 
action  of  the  spurs  would  be  negative. 

That  rolling  spurs  may  be  made  practical  in  place  of  TUN 
and  its  mate,  Fig.  140,  we  infer  from  the  curve  El',  that  instead 
of  going  outward  at  first  to  6rand  then  turning  in  toward  A  to  N, 
it  should  start  at  once  from  Ton  a  curvature  turning  toward  N. 
But  an  examination  of  the  case  will  readily  show  that  this  is  im- 
possible unless  the  arc  cd  is  increased. 

In  Fig.  141  the  problem  of  practicable  rolling  spurs  for  wheels 
A  and  B  is  undertaken,  where  NBV  is  the  total  angle  through 
which  B  must  turn  in  going  from  rest  to  engagement,  and  NAd' 
the  corresponding  angle  for  A. 

The  angle  NBV  is  divided  into  a  series  of  decreasing  angles 
from  N  toward  V,  then  the  same  number  of  ungbs  are  laid  off 
from  NA  toward  Ad,  all  of  which  span  slightly  more  arc  on  the 
radius  AN  than  does  the  first  of  the  series  .VFon  the  radius  BN. 
Then  the  curves  are  drawn  in  according  to  the  method  of  Fig.  69, 
giving  the  rolling  spurs  NUT  and  NO  V. 

It  is  plain  that  the  portions  of  the  rolling  curves  NG  and  NU 
are  so  nearly  perpendicular  to  the  radius  as  to  be  likely  to  work  bad 
in  practice  unless  provided  with  teeth,  becoming  part  of  the  toothed 
segment,  and  that  the  easement  curve  is  very  much  prolonged  as 
compared  with  that  of  Fig.  140.  Also  it  seems  that  the  speed  of 
B  in  some  cases  must  be  accelerated  beyond  that  of  the  gear  teeth 
in  engagement  and  then  retarded  to  the  latter  velocity. 

The  arc  of  engagement  e'd'  is  here  very  wide,  as  due  first  to  the 
very  considerable  drop  in  the  radius  from  AH  to  AN,  and,  second, 
to  the  fact  that  AN  is  so  short  compared  with  AB. 

It  appears,  then,  that  rolling  arcs  may  always  be  adopted  for 


120 


PRINCIPLES   OF   MECHANISM. 


engaging  or  disengaging  spurs  of  intermittent  wheels  in  place  of 

sliding  arcs  for  the  same;  but 
that  the  former  are  subject  to 
some  prejudicial  features  not 
inseparable  from  the  latter,  but 
which  outweigh  the  objection 
to  sliding  contact  action  of  the 
latter. 

These  wheels  may  be  made 
as  bevels  for  the  case  of  axes 
meeting,  a  complete  example 
in  metal  being  shown  in  Fig. 
98,  which  worked  satisfactorily. 
In  skew-bevels  they  are 
doubtless  possible,  but  proba- 
bly after  a  tedious  time  in 
construction  they  would  be 
found  unsatisfactory  mainly 
by  reason  of  the  sliding  along 
the  line  of  contact  of  the 
various  surfaces,  requiring  the 
spurs  to  be  very  thick,  etc. 

In  Fig.  140  the  wheel  A  is 
,  shown  with  only  one  locking- 
arc,  and  B  one  locking  seg- 
ment. But  either  may  have 
two  or  more,  involving,  how- 
ever, no  new  problems  over 
these  already  considered,  there 
being  a  toothed  arc  followed  by 
a  locking  arc,  and  the  latter 
again  by  a  toothed  arc,  etc.,  in 
succession,  spurs  being  provid- 
ed throughout. 

In  case  of  slow  movements 
of  this  kind  the  piece  FG  or 
S  in  Fig.  140  may  be  made 
shorter,  indeed  reduced  to  a 
mere  pin  as  at  E  and  H,  or  F 
FIG.  141.  and  />,  Fig.  141,  thus  simplify- 

ing the  movement  in  construction. 


NON-CIRCULAR   INTERMITTENT   MOTIONS.  121 

By  giving  the  spurs  a  very  considerable  curvature,  as  at  E  or  a 
Jf,  the  shock  due  to  contact  of  spur  and  pin  will  be  materially 
reduced  as  compared  with  that  attendant  upon  the  spurs  and  pins 
at  F  and  D.  Here  the  normal  EC  or  HCr  is  to  be  drawn,  and  the 
segments  ^16yand  1)C  measured  by  which  to  judge  of  the  shock  at 
contact  of  pin  and  spur.  The  shock  will  depend  upon  the  velocity- 

AC      AC' 

ratio  at  the  instant  of  initial  contact,  that  ratio  being  — —  or  —,777. 

u(u  u  L/ 

Comparing  with  Fig.  140,  it  seems  possible  that  pins  and  spurs  may 
be  so  made  that  the  shocks  are  no  more  prejudicial  than  in  Fig.  140. 


SOLID   ENGAGING   AND    DISENGAGING   SEGMENTS   OF 
INTERMITTENT   MOTIONS. 

Intermittent  motions  with  spurs  attached  are  objectionable  in 
•coarse  and  heavy  machinery  from  the  tendency  to  loosen  and  be- 
come detached.  These  wheels,  of  solid  casting  complete,  are  much 
the  more  satisfactory  in  such  places  as  binders  for  reaping  ma- 
chines, etc.,  both  for  simplicity  and  cost,  especially  where  they  are 
small  or  run  slowly.  By  suitable  shapes  of  the  engaging  and  dis- 
engaging segments,  however,  they  may  run  at  speeds  equalling 
those  provided  with  spurs,  as  in  Figures  140  and  141.  (See  Fig- 
ures 15  and  16.) 

A  suggestion  as  to  the  shapes  of  the  locking  segment  of  B  is 
obtained  from  Fig.  141  at  E  and  H,  where  the  spurs  might  seem- 
ingly be  in  the  plane  of  the  wheel  B,  and  A  correspondingly  cut 
away  to  a  circle  arc  coinciding  with  the  prongs  of  B  as  proposed, 
the  latter  so  connected  as  to  form  the  locking  segment  for  B. 
This  would  lead  us  at  once  to  Fig.  15.  Following  the  suggestion, 
we  obtain  Fig.  142  as  a  pair  of  these  wheels  of  the  non-circular 
form,  B  being  at  rest  on  the  locking  arc. 

In  designing  these  wheels,  the  distance  IL  and  LK  should  be 
sufficient  to  prevent  the  tendency  of  the  ends  of  the  prongs  /  and 
K  from  biting  into  the  arc  DLE  and  blocking  the  movement,  or,  in 
case  of  heavy  wheels,  causing  breakage.  To  examine  into  this,  we 
will  find  that  the  angle  BKM  must  exceed  the  so-called  "  angle  of 
repose  "  for  the  material  and  conditions  at  K.  A  portion  of  the  arc  at 
,/  should  be  cut  away  to  prevent  "  clinging  "  to  the  arc  IK  upon 
the  terminal  points  D  and  E  of  the  arc  DLE. 

Between  EG  and  DF  a  portion  is  cut  away  to  allow  the  prongs 


122 


PRINCIPLES   OF   MECHANISM. 


of  B  to  pass  as  the  latter  is  started  into  motion.  These  gaps  are 
undercut  at  G  and  F  to  reduce  shock  due  to  initial  contact  with, 
the  prongs  of  B,  the  intensity  of  which  shock  may,  as  before,  be 
examined  relative  to  the  distance  AP  or  AQ  from  the  center  of 
motion  of  A  to  the  normals  at  F  and  0.  This  shock  is  zero  when 
AP  and  A  Q  are  zero. 

No  direct  and  simple  rule  of  graphics  is  known  for  laying  out 
the  wheels  of  Fig.  142,  beyond  that  for  the  toothed  pitch  lines, 


FIG.  142. 


FIG.  143. 


where  the  arc  FNG  must  equal  RTS.  The  radius  AL  is  arbitrary,, 
except  that  it  ought  to  be  less  than  either  terminal  portion  of  the 
toothed  arc  FNHG.  After  assuming  a  trial  arc  DLE,  and  with 
trial  curves  for  the  wheel  A  at  F  and  G  sketched  in,  a  circle  may 
be  drawn  through  B  as  at  rack  for  the  latter  and  a  trial  templet  in 
cardboard  cut  out  by  guess  for  BIK,  with  a  hole  having  a  point  at 
B  for  center.  This  templet  with  guess  forms  at  //?  and  KS  is  to 
be  tried  at  several  positions  as  shown  at  EGH,  the  center  always 
remaining  on  the  circle  arc  through  B.  Likewise  at  FD.  A  guess 
shape  of  the  prong  of  B  at  /  may  be  tried,  and  if  found  unsuitable 
a  modification  suggested  by  this  trial  may  next  be  cut  out  and 
tried.  By  this  tentative  process  the  most  suitable  shapes  at  1R 
and  KS  and  at  DF  and  EGH  are  to  be  determined.  The  curves 
FHG  and  RTS  must  be  correct  rolling  arcs  for  gear  teeth.  Sev- 
eral trials  will  probably  be  needed  to  bring  out  the  most  satisfac- 
tory shapes  at  F,  I,  7f  and  G. 


ALTERNATE    MOTIONS.  123 

For  slow  movements,  where  the  shock  of  starting  and  stopping 
of  B  due  to  speed  is  at  a  minimum,  the  prongs  /  and  TTmay  be 
less  hooking  arid  simpler,  as  in  Fig.  143,  for  which,  at  a  given 
speed,  the  shock  is  about  four  times  as  great  as  for  Fig.  142. 

The  intensity  of  shock  may  be  examined  into  here  by  drawing 
the  normals  FP  and  GQ  and  considering  the  lengths  of  perpendic- 
ulars from  A  upon  these  normals.  These  are  seen  to  be  greater 
in  Fig.  143  than  in  Fig.  142  and  hence  the  greater  shocks. 

Bevel  wheels  after  either  variety,  Fig.  142  or  Fig.  143,  may  be 
readily  worked  out  according  to  Figs.  85  and  87  after  the  plain, 
wheels  are  drawn,  as  illustrated  by  Fig.  98  of  a 
working  model  in  brass.     Skew  bevels  are  here  v       ^^ 

much  more  practicable  than  in  the  wheels  of 
Figs.  140  and  141,  and  may  be  worked  out  ac- 
cording to  Figs.  93  to  95,  and  be  practical  in 
running  machinery. 

COUNTING   WHEELS. 

These  wheels,  of  high  durability,  may  be  con- 
structed as  in  Fig.  144,  which  is  a  positive 
movement  with  no  possible  likelihood  of  de- 
rangement except  by  breakage.  For  counting,  Fm  144 
the  upper  wheel  should  have  10  teeth.  The 
"  Geneva  stop"  is  a  familiar  example  of  this  class  of  wheels. 

ALTERNATE   MOTIONS. 
Directional  Relation  Changing. 

The  pitch  lines  for  these  wheels  have  been  treated  in  Figs.  9 9" 
to  103. 

The  most  essential  features  for  successful  wheels  of  this  kind 
consist  in  the  peculiar  shapes  of  terminal  teeth,  or  of  segments,  or 
of  spurs,  or  of  shifting  devices,  as  in  mangle  wheels. 

These  wheels  may  be  divided  into  two  distinct  classes,  viz.,  as 
those  admitting  of  limited  movements  only,  and  those  of  indefinite 
extent  of  movement,  to  be  taken  up  in  succession. 

FIRST.    LIMITED  ALTERNATE  MOTIONS. 

The  movement  is  in  this  case  limited  because  the  driven  piece 
must  unavoidably  make  a  forward  and  return  movement  for  each 
revolution  of  the  driver.  The  pitch  lines  of  these  wheels  have- 


124 


PRINCIPLES    OF    MECHANISM. 


-already  been  considered  in  Figs.  99  and  101,  and  the  shapes  of 
teeth  later. 

But  the  most  important  consideration  in  devising  these  move- 
ments is  that  they  shall  have  no  positions  for  skips,  derangements, 
etc.,  and  that  the  continued  action  is  absolutely  positive  through- 
out. Also  that  in  high  speeds  or  in  heavy  movements  of  this  kind 
the  driven  piece  should  be  given  a  retarding  motion  in  approaching 
the  stop,  and  an  accelerating  movement  when  it  moves  off  from  the 
stop  point,  to  avoid  shocks  and  breakages. 

Here,  as  in  the  intermittent  motions  of  Figs.  140  to  143,  there 
may  be  employed  engaging  and  disengaging  spurs  and  also  seg- 
ments. 

In  Fig.  145  we  have  an  example  where  spurs  and  pins  are  used 
in  which  A  is  the  driver  revolving  either  way  continuously,  and  B 


FIG.  145. 

the  follower  sliding  in  guides  D  and  E  forward  and  back  in  repeti- 
tion, a  complete  movement  both  ways  being  made  in  each  revolu- 
tion of  the  driver  A. 

The  wheel  A  and  rack  frame  B  are  of  the  same  thickness,  with 
teeth  to  engage  each  other  for  the  principal  movement,  the  correct 
construction  of  which  has  already  been  explained. 

In  the  wheel  A,  at  F  and  G,  are  pins  to  be  engaged  between  the 
spurs  FJ  and  GK  made  fast  to  the  side  or  sides  of  the  frame  and 
used  to  insure  the  transfer  of  engagement  of  teeth  from  the  one 
side  to  the  other  side  of  the  rack  frame  BB. 

The  spurs  are  here  made  to  conform  with  the  cycloidal  curves 
GL  and  PL  in  order  that  the  motion  of  B,  when  the  pins  move 
outward  in  their  slots,  may  be  such  as  to  allow  the  teeth  to  go  into 
engagement  without  interference.  The  spurs  F  and  G  may  be  ex- 


ALTERNATE    MOTIONS. 


125 


tended  as  far  as  desired  short  of  interfering  with  shaft  A,  but  the 
spurs  J  arid  JTmust  be  cut  away  so  as  to  allow  the  pins  to  enter 
the  slots,  just  as  they  necessarily  would  if  there  were  nothing  but 
pins  and  spurs.  In  this  example  J  and  K  are  cut  quite  abruptly,. 
K  most  so;  by  reason  of  which  quite  a  shock  will  occur  when  the 
pin  G  enters  its  slot  and  strikes  the  spur,  the  intensity  of  which 
may  be  judged  of  by  taking  account  of  the  length  of  the, perpen- 
dicular All  from  the  axis  A  to  the  normal  HG  to  the  spur.  The 
normal  FI  at  F  makes  the  perpendicular  distance  AI  relatively 
very  short,  from  which  it  appears  that  the  shock  due  to  contact  of 
the  pin  at  F  would  be  very  slight. 

To  reduce  the  shock  at  G,  the  spurs  may  be  extended  to  near 
M,  where  the  normal  to  the  cycloidal  curve  GML  passes  near  to  A. 
This,  however,  seriously  shortens  the  segment  FG  of  the  segmental 
wheel  A,  greatly  reducing  its  number  of  teeth.  The  movement 
would  thus  be  much  better  adapted  to  high  speeds,  and  judgment 
is  to  be  used  in  designing  to  obtain  the  best  adaptation  to  circum- 
stances. 

Assuming  the  velocity  of  A  constant,  the  velocity  of  movement 


FIG.  146. 

of  B  will  be  variable,  and  always  proportional  to  the  distance  on  a 
perpendicular  to  BB  from  A  to  the  intersection  with  the  normal 
to  the  spur  at  G  while  the  latter  bears  on  the  spur  and  to  the 
pitch  line  while  the  teeth  are  in  engagement. 

This  eccentric  form  of  A  for  alternate  motions  is  objectionable 
on  the  score  of  ease  of  movement,  and  may  be  avoided  except  where 
a  varied  motion  is  essential.  In  Fig.  146  the  variation  of  motion 
is  less  objectionable  than  here  for  a  durable  high-speed  movement. 


126.  PRINCIPLES   OF   MECHANISM. 

In  Fig.  146  we  have  an  example  where  solid  engaging  and  dis- 
engaging segments  instead  of  pins  and  spurs  are  adopted  with 
which  to  reverse  the  movement  with  certainty  and  positively.  As 
in  Figs.  142  and  143,  no  pieces  are  attached  by  screws  to  become 
loosened,  occasioning  derangement. 

At  reversal  of  motion  the  first  driving  contact  occurs  at  F  or 
G,  according  to  direction  of  motion.  Normals  at  the  contact  point 
being  FH  or  GI,  the  shock  due  to  contact  will  be  proportional  to 
the  length  AD  or  AE,  as  the  case  may  be,  DAE  being  perpendicu- 
lar to  the  direction  of  motion  of  BE  and  corresponding  to  the  line 
of  centers. 

The  contact  points  F  and  G  are  put  as  near  to  the  center  A  as 
possible  to  minimize  the  shock  due  to  initial  contact,  immedi- 
ately following  which  the  contact  moves  outward  to  the  pitch  line 
of  the  teeth  when  the  latter  engage  in  action  and  continue  the 
movement. 

This  movement  gives  the  driven  piece  B  the  most  rapid  motion 
when  near  the  middle  of  this  stroke,  the  rate  of  motion  being  for 
each  instant  always  proportional  to  the  length  of  the  perpendicular 
from  A  to  the  contact  on  the  pitch  line  for  that  instant. 

The  ends  of  the  first  teeth  of  both  A  and  B  are  cut  to  circle 
arcs  about  the  center  A  and  of  some  breadth  of  bearing,  to  prevent 
the  disabling  of  the  movement  unduly  soon  by  wear  of  those  parts. 

The  hub  of  A  is  made  to  fit  closely  into  the  extreme  portions 
of  the  opening  of  B  to  facilitate  bringing  A  and  B  into  the  correct 
relative  positions  when  the  contacts  between  teeth  are  shifted  from 
side  to  side. 

Any  non-circular  form  may  be  given  to  A,  as  preferred;  though 
it  will  be  found,  on  trial  of  several  various  curves,  that  it  is  favor- 
able for  smooth  running  to  have  both  the  curves  draw  in  toward 
A  as  in  Fig.  146,  instead  of  the  contrary,  as  in  the  case  of  one  side 
of  A  in  Fig.  145. 

These  wheels  may  be  made  with  conical  form  of  A  and  of  B, 
the  cones  having  a  common  vertex  more  or  less  remote  from  A,  and 
the  forward  and  back  motion  of  B  occurring  as  if  swinging  about 
an  axis  through  the  vertex  of  the  cone  and  perpendicular  to  the 
axis  of  A. 

Also  Figs.  145  and  146  may  be  laid  out  with  BB  on  a  circu- 
lar curve  and  the  same  be  put  into  the  spherical  form,  though  a 
considerable  cut  and  try  work,  with  templets,  will  probably  be  re- 
quired before  the  final  adopted  figures  are  brought  out. 


ALTERNATE   MOTIONS.  127 

The  Office  of  Motion  Templets. 

In  all  this  work  of  determining  wheels  and  curves,  as  in  Figs. 
140  to  146,  templets  will  be  found  most  useful.  They  may  be 
formed  of  paper  or  cardboard,  readily  cut  out  and  tried;  and  again 
others  cut  and  tried,  until  the  shapes  of  good  working  wheels  are 
arrived  at,  while  by  any  other  method,  coupled  with  contempt  for 
templets,  there  might  be  total  failure. 

SECOND.    UNLIMITED  ALTERNATE  MOTIONS. 
Mangle  Wheels  and  Hacks. 

The  pitch  lines  of  these  movements  have  been  considered  in 
Figs.  102  and  103,  and  the  forms  of  teeth  for  these  pitch  lines 
present  no  new  problems.  For  the  treatment  of  the  various  cases  of 
non-circular  pitch  lines,  and  teeth  for  the  same  which  are  likely  to 
arise  in  connection  with  this  subject,  we  may  refer  to  the  princi- 
ples already  given. 

For  a  mangle  rack  with  variable  velocity-ratio,  the  pinion  for 
the  same  may  be  non-circular  when  the  rack  pitch  line  is  to  be 
curved  as  in  Fig.  39  or  Fig.  46,  in  order  that  the  axis  of  the  pinion  A 
may  be  more  nearly  stationary  during  a  movement  forward  or 
back. 

For  a  mangle  wheel,  the  pinion  may  be  non-circular  and  the 
pitch  line  wavy  that  the  axis  may  follow  a  smooth  or  non-serrated 
groove,  as  in  Fig.  102,  or  103. 

The  pitch  line  on  the  mangle  wheel,  besides  being  wavy,  may  be 
varied  in  general  outline  from  smaller  to  greater  radius,  even  mak 
ing  several  turns  around  the  axis  of  the  driven  wheel  B  and  return 
to  the  starting-point.  In  this  case  the  velocity-ratio  will  vary  for 
the  non-circular  form  of  pinion,  A,  as  well  as  for  the  wandering  form 
of  pitch  line  on  the  wheel  B.  The  velocity-ratio  in  this  case  will 
consist  of  a  very  complicated  cycle  of  changes. 


CHAPTER  XII. 
TEETH   OF   CIRCULAR   GEARING. 

DIRECTIONAL  RELATION  CONSTANT.     VELOCITY-RATIO 
CONSTANT. 

FIRST:    AXES    PARALLEL. 

No  principles  in  the  theory  of  gearing,  beyond  those  already 
treated  under  non-circular  wheels,  remain  to  be  brought  out  here, 
the  leading  topic  for  our  present  study  under  circular  gears  being 
the  special  features  pertaining  to  circular  gearing,  and  some  prac- 
tical suggestions  concerning  them. 

Circular  gearing,  though  all  coming  under  one  general  theory 
in  common  with  non-circular,  viz.,  the  development  of  tooth  curves 
by  the  rolling  or  describing  templet,  may  yet  be  classified  and 
named  in  two,  and  possibly  three,  sections,  viz. : 

I.  Epicycloid al  Gearing. 
II.   Involute  Gearing. 
III.   Conjugate  Gearing. 

I.  EPICYCLOIDAL  GEARING. 

Here,  not  only  the  pitch  lines  are  circles,  but  the  describing 
circles  also,  so  that  the  tooth  outlines  are  epicycloidal  curves; 
described  by  a  tracing  point  in  the  circumference  of  a  rolling  circle, 
or  generatrix,  as  the  latter  rolls  on  the  periphery  of  the  pitch  circle 
as  directrix. 

When  the  rolling  is  on  the  outside  the  generated  curve  is  an 
epicycloid,  and  when  inside  it  is  an  hypocycloid  ;  tooth  curves  here- 
being  therefore  always  epicycloids  and  hypocycloids. 

128 


CIRCULAR   GEARING. 


129 


FIG.  147. 


SOME    PECULIAR    PROPERTIES  OF    EPICYCLOIDS  AND  HYPOCYCLOID&, 

In  Fig.  147  A  is  a  pitch  circle  upon  which  the  describing  circle 
D  rolls  carrying  the  tracing  point  P  and  tracing  the  curve  FPGH. 
This  curve,  thus  generated  by  the 
rolling  of  one  circle  upon  another, 
is  called  the  epicycloid.  If  the 
rolling  were  inside  the  directing 
circle  A,  as  in  Fig.  148,  it  is  called 
the  hypocycloid. 

One  peculiar  property  of  the 
epicycloid  is  that  it  may  be  gen- 
erated by  the  rolling  of  two  dif- 
ferent circles  upon  A;  viz.,  first, 
PD  rolling  at  /  with  the  tracing 
point  P  tracing  the  epicycloid 
FPGH-,  and,  second,  PE  rolling 
at  H  with  a  tracing  point  at  P, 
provided  that  the  diameter  of  PE,  less  the  diameter  of  PD,  equals 
the  diameter  of  A.  With  this  relation  of  diameters,  PD  =  AE, 
DA  =  PE,  and  triangles  PDI,  PHE,  and  1HA  are  all  similar  tri- 
angles. These  conditions  remaining  permanent  during  the  rolling, 
P  must  remain  on  the  line  through  /and  /fas  instantaneous  cen- 
ters of  motion,  so  that  PI  or  PH  remain  a  normal  to  the  epicycloid 
at  P  from  which  it  appears  that  one  and  the  same  epicycloid  FPGH 
will  be  described  by  either  circle,  rolling  as  described,  the  difference 
of  the  diameters  of  which  circles  equals 
that  of  the  directing  circle  A. 

Similarly,  if  two  circles  as  D  and  E 
roll  inside  of  the  pitch  line  A,  as  in  Fig. 
148,  the  sum  of  whose  diameters  equals 
that  of  the  pitch  line  A  and  carry  a  tracing 
point  P,  they  will  generate  one  and  the 
same  hypocycloid  FPG  ;  for  constantly 
PD  =  AE  and  DA  =  PE,  and  the  tri- 
angles IDP,  1AH,  and  PEH  are  all  mutu- 
ally similar,  and  hence  P  must  describe  the  same  hypocycloid  as 
stated. 

When  the  two  circles  D  and  E  are  equal,  as  in  Fig.  149,  the 
hypocycloid  becomes  a  straight  line  and  is  a  diameter  to  A,  as  in 
the  White's  parallel  motion. 


130 


PRINCIPLES    OF    MECHANISM. 


Epicycloidal  gearing  is  made  in  considerable  variety,  as  best 
adapted  to  various  purposes,  and  may  be 
classified  as  follows: 

1 .  Flanks  radial. 

2.  Flanks  concave. 

3.  Flanks  convex. 

4.  Interchangeable  sets. 

5.  "  Pin  gearing." 

6.  Rack  and  pinion. 

7.  Annular  wheels. 


F 

FIG.  149. 
Treating  these  separably,  we  have 


1st.  Flanks  Radial. 

In  Fig.  150,  we  have  a  pair  of  pitch  circles,  for  wheels  to  be 
fitted  with  epicycloidal  teeth  having  radial  flanks,  A  being  the 
driver,  B  the  follower,  and  C 
the  point  of  tangency. 

For  convenience,  we  will 
take  C  as  the  origin  of  all  the 
tooth  curves.  Then,  as  in  Fig. 
112,  the  describing  curve,  in 
this  case  the  circle  D,  is  placed 
inside  of  the  pitch  line  D  with 
the  tracing  point  a  at  C,  and 
rolled,  without  slipping,  to  the 
left  along  the  inside  of  B,  the 
tracing  point  a  describing  the 
tooth  curve  Ca.  Also  the 
circle  D'  of  the  same  diameter 
as  D  is  placed  on  the  outside 
of  A  with  its  tracing  point  b 
at  C,  and  rolled  toward  the 
left,  the  tracing  point  b  describing  the  tooth  curve  Ob.  The 
circles  D  and  D'  must  be  of  the  same  diameter,  or  may  be  the 
same  circle  and  tracing  point,  used  in  succession  inside  of  B  and 
outside  of  A.  The  tooth  curve  Ca  is  to  serve  as  a  flank  of  a  tooth 
for  B  and  the  curve  Cb  for  the  face  of  a  tooth  of  A,  the  two  mat- 
ing and  working  correctly  together  for  contacts  at  the  right  of 
ACB.  (See  Fig.  113  and  explanations.) 

In  like  manner  the  circles,  or  the  same  circle,  E  and  E'  are 
rolled  once  inside  of  A  and  once  outside  of  B,  with  the  tracer  c  or 


FIG.  150. 


CIRCULAR    GEARIKG.  131 

d  starting  at  C  and  tracing  the  mating  tooth  curves  Cc  and  Cd,  the 
former  for  a  flank  of  a  tooth  of  A  and  the  latter  for  a  face  of  a 
tooth  of  B,  the  two  furnishing  contacts  at  the  left  of  C,  all  as 
explained  in  Figs.  114  and  115. 

We  have,  now,  cCb  a  contiguous  face  and  flank  or  a  full  tooth 
curve  for  one  side  of  a  tooth  of  the  wheel  A,  and  likewise  aCd  the 
one  side  of  a  tooth  of  the  wheel  B,  which  teeth  mate  perfectly,  and 
will  work  together  smoothly  transmitting  motion  from  the  driver 
A  to  the  follower  B  with  a  constant  velocity-ratio  of  the  same  value 
as  if  the  directing  or  pitch  circles  A  and  B  were  rolling  one  upon 
the  other  without  slipping. 

The  pitch  lines  being  here  circles,  other  tooth  curves  described 
with  the  same  describing  circles  will  generate  tooth  curves  which 
will  be  copies  of  those  already  obtained,  as  cCb  and  aCd.  Hence 
it  is  only  necessary  to  copy  these  curves  as  often  as  needed  at  the 
proper  pitch  spaces  around  the  wheels,  when,  on  describing  the 
addendum  and  dedendum  circles,  the  drawings  of  the  wheel  teeth 
will  be  completed,  fillets  being  struck  in  as  hereafter  explained. 

The  leading  peculiarity  of  these  teeth  is  that  they  have  radial 
flanks  due  to  the  fact  explained  in  Fig.  149  that  the  describing 
circles  have  diameters  equal  half  those  of  the  pitch  circles  within 
which  they  roll,  and  hence  the  styling  radial  flanks  for  these 
wheels. 

Following  the  contact  between  these  teeth  from  beginning  to 
end,  for  A  turning  right-handed  driving  B,  we  find  that  the  initial 
contact  between  a  pair  of  teeth  occurs  inside  of  the  pitch  circle  A 
to  the  left  of  C,  and  moves  on  through  the  point  C  and  ends  inside 
of  the  pitch  circle  B  at  the  right  of  (7,  and  following  along  an  "  S  " 
shaped  curve;  for  full  discussion  of  which,  see  path  of  contact,  limit 
of  contact,  etc.,  Fig.  177. 

For  convenience  in  the  drawing  of  radial  flanked  teeth,  we  see 
from  the  above  that  it  is  only  necessary  to  draw  flanks  as  radial 
lines,  and  faces  by  using  a  rolling  circle  on  the  outsides  of  a  pitch 
line,  the  diameter  of  which  equals  the  radius  of  the  mating  pitch 
circle. 

For  proportions  of  teeth,  see  Fig.  110  and  rules  accompanying. 


132 


PRINCIPLES   OF    MECHANISM. 


FlG.  151. 


2d.  Flanks  Concave. 

In  Fig.  151  we  have  a  pair  of  pitch  lines,  A  and  B,  to  be  equipped 

with  epicycloidal  teeth  which  have  con- 
cave flanks,  the  object  of  which  may 
be  to  give  to  the  teeth  greater  strength 
than  if  the  flanks  were  radial.  The 
increased  thickening  of  teeth  in  the 
flanks  may  be  judged  by  noting  the 
departure  of  the  hypocycloid  from  the 
radius  in  Fig.  151. 

The  size  of  the  describing  circle  is 
arbitrary,  and  may  be  chosen  to  meet 
the  most  fastidious  taste  or  fancy  as 
to  thickness  or  strength  of  tooth,  the 
only  restriction  being  that  the  diame- 
ters of  the  circles  D  and  D'  be  the 
same,  and  likewise  for  E  and  Ef . 

The  complete  tooth  curve  for  A, 
and  also  for  B,  is  shaded  to  distinguish  it.  A  templet  may  be 
fitted  to  this  curve  and  mounted  to  swing  around  the  center,"  and  at 
every  point  for  a  tooth  curve  it  may  be  held  and  the  tooth  curve 
marked  or  copied  from  the  templet,  as  explained  among  practical 
operations.  (See  Page  153.) 

With  the  tooth  curves  marked  off  around  the  wheels,  the  ad- 
dendum and  dedendum  circles  drawn  in,  and  the  root  fillets  struck, 
the  tooth  outlines  for  the  wheels  will  be  completed. 

3d.  Flanks  Convex. 

In  Fig.  152  the  rolling  circles  have  diameters  which  are  greater 
than  the  radii  of  the  wheels  within  which 
they  roll,  as  in  the  case  of  the  wheel  E  in 
Fig.  128,  or  /,  Fig.  119,  giving  flanks 
of  teeth  that  are  convex.  The  finished 
teeth  for  this  case  will  be  found  very 
weak,  as  evident  from  the  undercut  ap- 
pearance which  the  teeth  present.  The 
filleting  at  the  root  will  help  this,  though 
the  teeth  are  still  weaker  than  in  the 
previous  cases,  but  will  be  found  of 
favorable  form  in  light-running  machine 
ery.  FIG.  152. 


CIRCULAR   GEARING. 


133 


4th.  Interchangeable  Sets  of  Gears. 

These  wheels  are  called  interchangeable  because,  of  any  number  of 
them  of  various  sizes  having  the  same  pitch  and  describing  circle, 
any  two  will  work  together  correctly.  This  is  seen  not  to  be  the 
case  with  Figs.  150  to  15:2. 

The  essential  requirement  for  this  in- 
terchangeability  is  simply  that  the  gener- 
ating circles  D,  D',  E,  and  Ef  be  always  of 
the  same  diameter  for  generating  faces  and 
flanks  of  teeth  throughout,  as  shown  in 
the  circles  D,  D,  etc.,  in  Fig.  153. 

For  instance,  A  and  B  is  a  pair  of 
wheels  differing  in  size  and  selected  at 
random  from  the  set,  neither  being  the 
largest  nor  smallest,  for  which  we  have 
a  pair  of  tooth  curves  in  contact  at  (7, 
shaded  as  shown,  the  one  a-Cd  belonging 
to  wheel  A,  another  bCc  to  wheel  B,  etc. 

Now,  the  curve  aCd  is  throughout  de- 
scribed by  the  rolling  circle  D,  and,  as  the 
latter  is  chosen  independently  of  BB', 
etc.,  it  appears  that  this  tooth  curve  aCd 
is  entirely  free  of  all  reference  to  any  other 
wheel  of  the  set,  and  only  dependent  upon 
the  pitch  line  A  and  describing  circle  D.  The  same  is  true  of  any 
other  wheel  of  the  set,  as  A',  B,  B',  etc.,  the  tooth  curve  of  any 
one  wheel  being  peculiar  to  that  wheel  only,  since  the  describing 
circle  is  one  and  the  same  throughout  the  set. 

Hence,  to  realize  an  interchangeable  set  of  epicycloidal  gear 
wheels,  where  any  two  whatever  will  mate  correctly,  it  is  only 
necessary,  in  addition  to  a  constant  pitch,  that  the  describing  circle, 
generating  the  teeth,  be  one  and  the  same  throughout.  The 
"  change  wheels"  of  engine  lathes  in  machine  shops  is  a  well-known 
example  of  gears  of  this  kind ;  that  is,  in  "  sets." 

In  Fig.  153,  three  wheels  of  a  set  of  this  kind  are  shown  with 
tooth  curves  drawn  where  the  describing  circle  is  in  common  for  all. 

At  the  size  B'  the  flanks  are  radial,  as  in  Fig.  150,  because  here 
we  strike  the  size  where  the  radius  of  B'  equals  the  diameter  of  the 
describing  circle  D.  For  wheels  of  this  set  smaller  than  B'  the 
flanks  would  be  convex,  as  in  Fig.  152. 


FIG.  153. 


134  PRINCIPLES   OF   MECHANISM. 

As  the  wheels  of  the  set  are  made  larger,  that  one  with  infinite 
radius  becomes  a  rack,  where  the  curves  for  faces  are  identical  with 
those  for  flanks. 

If  we  carry  the  variation  of  curvature  still  further  in  the  same 
direction,  the  rim  of  the  wheel  becomes  concave,  and  the  smaller 
wheel  of  the  pair  finds  its  place  inside  the  larger.  This  is  some- 
times called  internal  or  annular  gearing.  In  this  case  a  given  an- 
nular wheel  has  tooth  curves  which  are  exactly  the  same  as  those 
for  a  wheel  of  the  same  diameter  with  its  mating  gear  in  outside 
action. 

In  the  well-known  example  of  the  "change  wheels"  of  engine 
lathes,  the  gears  and  teeth  are  small,  and  correct  teeth  are  not  so 


FIG.  154. 

important  as  in  sets  of  patterns  for  cast-iron  mill  gearing  with 
larger  teeth,  where  the  manufacturer  must  be  prepared  to  furnish 
pairs  of  gears  of  almost  any  sizes  with  correct  teeth.  The  best  way 
to  meet  this  demand  is  to  have  patterns  made  in  sets,  as  above  ex- 
plained, any  two  of  which  will  mate  correctly;  then  with  all  of  this 
set  an  occasional  new  pattern  will  work  correctly,  one  new  pattern 
thus  providing  for  many  new  pairs. 

5th.  Pin  Gearing. 

This  is  called  pin  gearing  because  the  teeth  of  one  wheel  are  pins 
and  of  the  other  are  spurs  to  engage  the  pins,  as  shown  in  Fig.  154. 

To  draw  correct  working  gearing  of  this  kind,  take  A  and  B  as 
the  pitch  lines,  and  the  circle  at  C as  the  section  of  a  cylindrical  pin 
or  tooth.  Draw  an  epicycloid  CD,  with  B  as  the  describing  circle. 


CIRCULAR   GEARING. 


135 


This  epicycloid  will  be  the  path  of  the  center  of  the  pin  (7,  on  the 
supposition  that  the  pin  is  made  fast  to  the  circle  B,  and  that  the 
latter  is  rolled,  without  slipping,  along  the  pitch  line  A,  as  shown. 
The  circles  along  the  epicycloid  CD  represent  the  pin  tooth  in 
various  positions  agreeably  with  corresponding  positions  of  the 
pitch  line  B,  in  its  rolling,  and  the  curve  ab,  just  tangent  to  these 
circles  of  the  pin,  will  be  found  a  correct  tooth  curve  for  one  side 
of  a  tooth  or  spur  of  the  wheel  A,  by  aid  of  which  all  the  teeth  of 
the  wheel  A  may  be  drawn.  The  pair  of  gear  wheels  A  and  B  may 
therefore  be  fully  drawn,  the  pins  and  teeth  being  properly  dis- 
tributed in  pitch. 

A  brief  examination  of  Fig.  155  will  show  that  the  contacts 
between  these  teeth  and  spurs  are  on  one  side  of  the  line  of  centers 
AB,  receding  if  A  is  driver,  and  approaching  if  B  is  driver. 
Also,  it  is  found  that  the  receding  contacts 
are  the  most  efficient,  as  clearly  evinced  by 
the  extreme  case  of  Fig.  156,  where,  with  A 
for  driver,  the  extended  tooth  curve  acts 
like  a  wedge  to  force  the  pin  along  in  its 
path,  while  if  B  is  driver  the  pin  is  forced 
down  upon  the  tooth  curve  of  A,  the  latter 
serving  very  much  to  block  or  to  hinder  the 
movement  of  B\  and  when,  as  in  practice, 
friction  between  the  pin  and  tooth  is  con- 
sidered, the  problem  is  still  more  prejudiced.  FIG.  155. 
Fig.  156  is  an  extreme  case  to  illustrate  more  forcibly,  but  in  a  more 
moderate  one,  as  in  Fig.  155,  the  principle  still  holds  to  a  certain 
extent. 

As  tooth   contacts   are  on  one  side  of  the 
line  of  centers,  the  arc  of  contact  is  consider- 
ably limited,  and  each  problem  should  be  exam- 
ined with  care  to  determine  whether  one  pair 
of  teeth  quit  contact  before  the  next  succeeding 
pair  commence  it,  the  latter  usually  occurring 
when  the  center  of  the  pin  is  some  considerable 
FIG.  156.          distance  past  the  line  of  centers.     (See  Mac- 
Cord 's  Kinematics,  page  210.)     For  large  pins,  the  amount  to  allow 
for    this   is   greatest,  and  vanishes   as  the  pin  diameter  becomes 
zero. 

In  some  examples  these  pin  teeth  have  been  made  as  rollers  on 
smaller  pins,  with  the  view  of  reducing  the  friction. 


136 


PRINCIPLES    OF   MECHANISM. 


Pin  gearing  has  been  quite  extensively  used  in  the  cheaper 
brass  clocks  which  have  flooded  the  country  within  the  last  forty 
years.  In  these  the  wheels  are  drivers,  as  seen  above  to  be 
advisable,  and  made  of  sheet  brass,  while  the  pinions  have  pins  of 
straight  and  smooth  steel  wire,  held  at  both  ends  in  holes  in  a  pair 
of  heads  or  disks  of  brass. 

Inside  Pin  Gearing. 

Pin  gearing  is  applicable  in  inside  gearing,  the  curve  CD,  Fig. 
154,  instead  of  being  outside  would  be  inside,  and  consequently  an 
hypocycloid,  but  the  same  principles  apply  as  before. 

In  the  special  case  for  inside  gearing,  where  the  pinion  is  half 
the  size  of  the  wheel,  the  pins  of  the 
pinions  move  along  certain  diameters  of 
the  wheel,  according  to  Fig.  149,  and 
sliding  blocks  in  grooves  may  be  intro- 
duced, with  holes  to  receive  the  pins  of 
the  pinion  as  in  Fig.  157. 

One  of  the  arms  of  B  may  be  cut 
away  and  a  grooved  arm  of  A  also,  and 
still  the  movement  would  work  complete. 
.The  remaining  two  arms  of  B  could 
be  placed  at  any  other  angle  with  each 
other  as  45°,  87°,  etc.,  provided  the 
corresponding  two  grooved  arms  of  A 
were  at  half  the  angle  intervening, 
and  the  action  would  be  perfect. 

Pinion  of  Two  Teeth. 

Approximately,  rectangular  teeth 
may  be  employed  instead  of  cylindric, 
as  shown  in  Fig.  15-8,  where  the  longer 
sides  of  the  pins  are  radial  with  B 
when,  as  shown  by  Fig.  149,  the 
circle  BEF  as  a  templet  rolling  on 
the  circle  AF,  a  tracing  point  at  E 
would  describe  the  epicycloid  KDE, 
while  if  rolled  inside  of  BF  it  would 
describe  a  radial  side  of  the  tooth  or 
pin  E  for  the  curve  KDE  to  work 
against. 


FIG.  157. 


FIG.  158. 


CIRCULAR   GEARING. 


137 


A  is  here  the  pinion  and  has  only  two  teeth,  the  least  number 
possible,  though  it  may  have  as  many  more  as  desired. 

The  construction  is  a  little  peculiar,  the  teeth  of  B  alternating 
upon  opposite  sides  of  a  central  disk,  and  for  A  we  have  two 
heart-shaped  pieces,  KDEGH  and  FJID,  at  such  distance  apart 
on  a  hub  that  the  disk  of  B  may  work  between.  Then  the  con- 
tacts will  alternate  from  side  to  side  as  the  rotation  proceeds. 
The  part  AEG  will  just  fill  the  space  between  the  two  teeth  E 
and  G,  and  the  wheels  may  work  either  way. 

It  is  hardly  practical  to  make  B  driver  here,  since  the  common 
normal  at  E  passes  so  near  to  A  that,  considering  friction,  A  would 
serve  as  a  block  to  hinder  the  driving  action  of  B. 

In  Fig.  159  we  have  a  two-leaved 
pinion  A  driving  B  as  equipped  with 
cylindric  pins,  the  construction  be- 
ing the  same  as  in  Fig.  154,  except 
carried  to  the  extreme  case  of  two 
teeth  for  A. 

Care  should  be  exercised  that  a 
beginning  contact  at  D  is  fairly 
made  by  the  time  a  preceding  con- 
tact ceases  at  E.  The  rolling  of  the 
pitch  line  B  as  describing  templet^ 
on  the  pitch  line  A  C,  gives  rise  to 
the  epicycloid  dotted  in  at  FG, 
parallel  to  which  and  at  a  pin  radius 
EG  from  it  is  drawn  the  tooth 
curve  for  the  pinion  A  as  shown,  as 
in  Fig.  154. 

For  inside  gearing,  the  same  pinion  A  may  be  placed  inside  the 
same  wheel  B,  when  the  pitch  line  of  B  will  fall  at  HI,  tangent  to 
A\  and  the  pins  will  have  contacts  at  //and  /as  shown,  the  center 
of  B  being  at  B',  and  below  B  by  the  amount  2 AC. 

Rack  and  Pinion. 

For  the  case  of  the  rack  and  pinion,  the  latter  may  have  either 
the  pins  or  the  teeth,  the  same  principles  applying. 

Wlieels  of  Least  Crowding  and  Friction. 

A"  peculiar  form  of  pin  gearing  is  obtained  by  use  of  a  describing 
circle  of  somewhat  unusual  size,  as  in  Fig.  160,  the  pin  having  its 
side,  which  is  within  the  pitch  circle,  a  complete  hypocycloid. 


FIG.  159. 


138 


PRINCIPLES   OF   MECHANISM. 


Here  the  common  normal  makes  the  least  possible  maximum 
obliquity  with  the  common  tangent  to  the  pitch  lines,  when  the 


FIG.  160. 

teeth  are  made  just  so  large  that  the  arc  of  contact  embraces  but 
one  tooth  contact  at  once;  this  obliquity  being  but  about  half  that 
for  circular  pins  when  the  contacts  must  all  be  on  one  and  the  same 
side  of  the  line  of  centers,  as  in  Fig.  154,  and  also  only  about  half 
the  obliquity  of  teeth  in  Fig.  150. 

These  wheels  of  all  others  will  have  the  least  possible  crowding- 
action  or  tendency  to  force  the  axes  from  each  other  when  made  as 
just  specified,  so  that  the  arc  of  contact  but  slightly  exceeds  the 
pitch,  the  friction  due  to  crowding  being  therefore  at  a  minimum, 
and  the  pressure  to  turn  the  driven  wheel  will  be  at  a  minimum 
for  two  reasons:  first,  because  the  driving  pressure  between  teeth  is 
nearly  tangent  to  the  driven  wheel,  and,  second,  because  this  last- 
named  pressure  is  to  be  compounded  with  the  least  possible  crowd- 
ing pressure. 

But  the  teeth  of  these  wheels  are  unusually  weak,  and  it  can  be 
recommended  for  only  light-running  machinery. 


CIRCULAR    GEARING. 


139 


F 

FIG.  161. 


On  account  of  the  slight  obliquity  of  the  line  of  action  with  the 
common  tangent  to  the  pitch  lines,  a  variation  of  the  distance 
between  the  centers  A  and  B  would  entail  less  prejudicial  action 
than  in  Figs.  150  to  159,  so  that  in  Fig.  160  we  find  a  suggestion 
for  such  wheels  as  used  on  clothes-wringers,  straw-cutters,  etc., 
when  the  distance  between  A  and  B  is  varied  in  a  very  large  ratio. 

6th.    The  Rack  and  Pinion. 

In  any  of  the  preceding  cases  for  Figs.  150  to  160,  we  obtain  a 
rack  and  pinion  by  giving  to  one  of  the  wheels  of  the  pair  an  in- 
finite radius. 

For  the  case  of  radial  flanks,  as  in 
Fig.  150,  if  we  make  A  infinite,  we  ob- 
tain Fig.  161. 

Here  the  radial  flank  of  A  will  be 
simply  a  perpendicular  to  the  straight 
line  A  C,  while  the  face  curve  CD  for  B 
will  in  this  particular  case  be  an  involute  \  \f 
to  B,  since  the  describing  circle  to  roll 
inside  of  A,  which  is  infinite,  will  be  in- 
finite also;  ED  being  one  position  of  that  circle  or  straight  line, 
carrying  the  tracer  D,  and  tracing  the  involute,  CD,  as  the  rolling 
advances,  along  CE.  Thus  the  line  DE  will  equal  the  arc  CE9 
and  the  involute  tooth  face  CD  can  readily  be  drawn. 

For  the  radial  flank  of  B  and  face  of  A  the  describing  circle 
GH  must  be  half  as  large  as  B,  which,  with  a  tracer  H,  rolled  on 
A  C  will  in  this  particular  case  trace  a  cycloid,  Off. 

Then  we  will  have  FCH  for  a  full  tooth  curve  for  A,  and  BCD 
a  full  tooth  curve  for  B;  and  the  correct  teeth  can  be  constructed, 

As  before,  CD  and  CF  will  be  mat- 
ing tooth  curves,  having  a  working* 
contact  to  the  right  of  C,  while  the 
mating  tooth  curves  CB  and  CH  will 
have  contact  at  the  left  of  C. 

One  peculiarity  of  the  action  be- 
tween CD  and  OF  is  that  only  the  point 
C  of  the  line  CF  will  go  into  action 
with  the  whole  of  CD,  and,  as  a  con- 
sequence, the  rack  teeth  would,  in  prac- 
tice, become  unduly  worn  at  C. 


FIG.  162. 


140 


PRINCIPLES    OF    MECHANISM. 


For  the  case  of  concave  flanks,  the  describing  circle  G,  Fig.  162, 
must  have  a  diameter  less  than  the  radius  of  B  to  describe  the 
mating  tooth  curves  CI  and  CH ";  and  any  circle,  J,  less  than 
infinity  will  serve  to  describe  the  mating  tooth  curves  CD  and  CF. 
The  line  FCH  is  a  tooth  profile  for  the  rack  AC,  and  the  line  DC1 
is  a  tooth  profile  for  the  pinion  B.  With  these  the  teeth  can  be 
fully  drawn  for  the  rack  and  pinion. 

The  curves  CFand  CH  will  be  cycloids,with  bases  on  the  line  A  C. 

For  the  case  of  flanks  convex,  the  describing  circle,  as  in  Fig. 
152,  to  roll  inside  of  B,  must  have  a  diameter  greater  than  the 
radius  of  B\  but  for  the  rack,  convex  cycloidal  flanks  are  impossible 
for  the  reason  that  the  straight- line  flank  of  Fig.  161  is  struck  with 
-an  infinite  circle,  the  largest  possible.  Also,  it  is  seen  that  by  roll- 
ing any  describing  circle  along  CA,  from  C  towards  A,  the  line 
CF  could  not  be  described  as  downward  toward  the  right,  for  the 
reason  that  no  instantaneous  center  on  CA  will  serve  for  striking  a 
curve  in  such  position.  The  straight  line  CF,  normal  to  CA, 
Fig.  161,  is  therefore  the  extreme  case  of  carrying  F  toward  the 
right. 

The  Case  of  the  Rack  in  Inter 'changeable  Sets  has  already  been 
treated  in  connection  with  Fig.  153,  also  Pin  Gearing,  in  connec- 
tion with  Figs.  155  and  159. 

7th.  Annular  Wheels. 

In  Fig.  163  we  have  the  general  case,  the  smaller  pitch  line 
being  within  that  of  the  annular  wheel.  t  The  circle  carrying  the 

tracers  If  or  I  is  rolled  inside  of  A,  and 
also  inside  of  B,  to  generate  u  face  of 
a  tooth  of  A  and  a  flank  of  B.  These 
curves  are  both  hypocycloids;  but,  ac- 
cording to  the  general  theory,  are  seen 
to  be  the  proper  ones  to  work  together 
for  a  face  and  flank.  For  the  outside 
curves  the  circle  J  carries  the  tracer 
D  or  F  to  describe  the  epicycloids  CF 
and  CD  as  mating  curves  for  a  face  of  B 
and  flank  of  A. 

Both  sets  of  flanks  are  here  concave, 
and  that  for  A  must  always  be  so; 
though  that  for  B  may  be  radial  or  convex  by  using  a  larger  de- 
scrioing  circle  to  carry  the  tracers  H  and  /. 


FIG.  163. 


CIRCULAR    GEARING. 


141 


A  peculiar  case  of  interference  of  the  teeth  of  the  annular  wheel 
and  its  pinion  occurs  when  the  sum  of  the  diameters  of  the  describ- 
ing circles  G  and  J  exceeds  the  difference  of  diameters  of  A  and 
By  as  discovered  by  A.  K.  Mansfield  and  Prof.  C.  W.  MacCord. 

The  limit  occurs  when  the  added  diameters  of  the  describing 
circles  G  and  J  just  equals  the  difference  of  diameters  of  the  pitch 
lines  A  and  B,  as  shown  in  Fig.  164.  At  this  limit  the  faces  of  the 
teeth  of  the  internal  gear  A  have  correct  acting  tooth  contacts  with 
the  faces  of  the  teeth  of  the  in- 
side pinion  B,  as  may  be  proved 
by  aid  of  the  alternate  describ- 
ing circle  K. 

Referring  to  Fig.  148,  it  is  seen 
that  the  hypocycloid  DL,  Fig. 
164,  may  be  regarded  as  being  now 
traced  by  either  G  or  K  carry- 
ing the  tracer  /);  and  that  aDC 
is  a  straight  line.  Also  referring 
to  Fig.  147,  it  is  clear  that  the 
epicycloid  DM  may  be  regarded 
as  in  the  act  of  being  traced  by 
either  J  or  K  carrying  the  tracer 

D,  and  that  the  points  DbCure  in  a  straight  line.  Hence, 
points  in  one  and  the  same  straight  line;  and  the  hypocycloid  DL 
and  epicycloid  DM  are  tangent  to  each  other  at  D,  as  required 
for  tooth  contacts;  and  we  have,  at  this  limit,  in  addition  to  the 
usual  face  and  flank  contacts,  this  face  to  face  contact  at  D. 

If  D  -f  J  is  larger  there  will  be  interference  at  D  between  faces, 
while  if  smaller  this  face  to  face  contact  is  lost. 

In  this  particular  case  we  have  the  advantage  over  ordinary 
gears  of  an  additional  and  unusual  contact  at  D,  and  a  most  excel- 
lent one,  except  for  its  remoteness  from  C,  where  the  amount  of 
slip  of  face  surfaces  over  each  other  is  greater  than  in  the  other 
and  usual  contacts. 

From  wThat  has  been  said,  it  is  plain  that  the  describing  circles 
G  and  J  can  be  varied,  either  smaller  than  the  other,  but  that  they 
must  not  be  larger  than  that  they  will  fit  in  between  the  two  pitch 
circles,  as  shown,  under  penalty  of  positive  interference  of  the  teeth 
at  D,  but  that  they  may  be  smaller  than  shown,  with  loss  of  con- 
tacts at  D. 

WJieels  in  sets  may  be  worked  in  annular  gearing,  that  is,  any 


FIG.  164. 


142  PRINCIPLES    OF    MECHANISM. 

of  a  number  of  pinions  may  work  correctly  in  any  of  a  number  of 
annular  wheels,  provided  that  one  and  the  same  describing  circle 
be  used  throughout,  and  that  the  facts  of  interference  be  observed, 
as  pointed  out  above. 

Pin  annular  gearing  may  also  be  used  by  following  the  princi- 
ples explained  under  pin  gearing,  and  the  pins  may  be  in  either  the 
wheel  or  pinion. 

II.  INVOLUTE  GEARING, 

As  explained  in  connection  with  Figs.  119  and  121,  logarithmic 
spirals  may  be  employed  for  describing  curves  which,  rolled  on  the 
pitch  lines  A  and  B,  inside  and  outside  in  the  usual  way,  will  de- 
velop faces  and  flanks  by  pairs,  the  curves  thus  traced  being  invo- 
lutes to  circles  somewhat  smaller  than  the  pitch  lines. 

Thus,  in  Fig.  165,  take  the  pitch  line  A—DCH  and  roll  the 
logarithmic  spiral  FD  from  C  as  shown,  with  a 
tracer  at  the  pole  F.  The  curve  CF  will  be  an 
involute  to  a  circle  EG,  inside  of  A,  to  which  the 
line  EDF  is  a  tangent.  This  is  made  evident 
by  the  fact  that  the  logarithmic  spiral  is  a  curve 
of  constant  obliquity,  that  is,  that  any  line  or 
radius  vector,  FD,  makes  a  constant  angle  with 
a  tangent  to  the  spiral  at  D,  and,  consequently, 
FIG.  165.  a  constant  angle  with  the  tangent  to  the  circle  at 

D,  and  also  with  a  radius  to  the  circle  at  D.  Hence,  FDE  will  al- 
ways be  tangent  to  some  one  circle  EGA ;  and  it  is  plain  that  we 
will  obtain  one  and  the  same  curve,  CF,  whether  the  latter  be  gen- 
erated by  the  rolling  of  the  spiral,  with  tracer  F,  along  CD,  or  by 
the  unwinding  of  the  cord  FE,  with  tracer  F,  from  the  circle  GE, 
This  curve,  CF,  will  thus  be  an  involute  to  the  circle  AGE.  This 
curve  as  an  involute  starts  at  G,  while  as  developed  by  FD  it 
seems  to  start  at  C.  To  supply  CG  by  the  rolling  spiral,  place  the 
pole  and  tracer  at  C,  and  roll  inside  along  CH  as  shown,  when  CG 
will  be  traced  just  to  G  and  stop;  for  then  the  radius  of  curvature 
of  the  spiral  at  #will  be  HA,  since  to  construct  this  radius  of 
curvature  for  a  logarithmic  spiral  draw  the  right-angled  triangle 
A  GH,  when  AH  is  that  radius.  The  spiral  then  cannot  roll  beyond 
H,  and  the  curve  CG  stops  at  G. 

As  tooth  curves,  CF  is  to  serve  as  a  face  and  CG  as  a  flank,  and, 
as  developed  by  the  spiral,  are  seen  to  fall  within  the  general  theory 
of  development  by  rolling  curves.  But,  in  practice,  for  convenience, 


CIRCULAR    GEARING. 


143 


the  curves  are  drawn  as  involutes  to  the  circle  GE,  called  the  base 
circle  or  base  of  the  involutes. 

When  the  teeth  need  to  be  cut  deeper  between  than  CG,  such 
portion  is  cut  away  arbitrarily  below  EG  as  shall  make  room  for 
the  interengaging  teeth. 

To  Obtain  Tooth  Curves  for  Wheels  with  Involute  Teeth. — Draw 
the  pitch  circles  A  and  B  in  common  tangency  at  C,  Fig.  166, 
then  a  straight  line  D  CE  through  the  point  of  common  tangency 
or  pitch  point  C,  and  perpendic- 
ulars AD  and  BE  from  the  cen- 
ters A  and  B  upon  this  line. 
Then,  with  AD  and  BE  as  radii, 
draw  the  base  circles  FD  and  HE 
of  the  involutes.  The  involute 
may  be  drawn  by  unwinding  a 
thread  from  the  base  circle,  the 
thread  having  the  tracer  attached, 
or  by  drawing  several"  tangent 
lines  as  shown,  and  stepping  with 
dividers  from  C  to  D,  then  with 
the  same  number  of  steps  to  Fy 
then  back  to  other  points  of 
tangency  and  out  on  the  tangent 
to  points  G,  etc.,  for  as  many 
points  as  desired,  when  the  invo- 
lute FOG  may  be  drawn  in;  likewise  for  HCI. 

This  gives  us  a  pair  of  tooth  curves  from  which  the  teeth  may 
be  drawn  in,  clearance  room  within  the  base  circles  DF  and  EH 
being  assumed.  As  the  finished  gear  wheels  revolve,  the  tooth 
contacts  all  occur  on  the  line  DE,  remaining  on  and  moving 
along  it;  and  there  can  be  no  correct  contacts  outside  of  the 
length  DE,  though  between  D  and  E  there  may  be  several.  At- 
tempted contacts  beyond  D  or  E  give  rise  to  serious  interference. 

In  practice  it  is  rare  that  the  contacts  extend  to  the  whole  of 
DE,  though  they  may,  by  laying  out  the  wheel  with  that  result  in 
view,  as  by  making  the  addendum  circles  cut  at  D  and  E. 

These  gear  wheels  possess  the  peculiar  property  that  the  dis- 
tance between  centers  A  and  B  may  be  varied  at  pleasure,  without 
altering  the  correctness  of  action  of  the  teeth  upon  each  other. 

For,  examining  Fig.  166,  we  find  that  the  triangles  A  CD  and 
BCE  are  similar,  so  that  the  pitch  point  C  is  always  at  the  same 


FIG 


144 


PRINCIPLES    OF    MECHANISM. 


proportional  position  between  A  and  B9  whatever  the  magnitude 
of  AB,  allowing  AD  and  BE  to  remain  constant.  Under  these 
conditions  any  number  of  correct  working  tooth  outlines  may  be 
drawn  with  varied  distances  AB.  The  spaces  between  the  teeth 
along  on  the  line  FD,  or  on  DE,  or  on  HE  all  equal  the  normal 
pitch,  which  is  constant;  while  the  spaces  between  the  teeth  along 
on  the  pitch  circles  equal  the  circumferential  pitch.  This  latter  dif- 
fers slightly  with  the  normal  pitch,  and  is  slightly  variable  if  we 
regard  the  pitch  circles  as  remaining  tangent  to  each  other  at  0 
while  the  line  of  centers  AB  varies. 

This  gearing  is  often  recommended  for  situations  where  the 
distance  AB,  through  carelessness,  wear  of  parts  or  otherwise  is 
subject  to  changes  ;  and  where  epicycloidal  gearing,  requiring  A B 
to  remain  constant,  will  not  serve  correctly. 

These  involute  wheels  in  a  group  of  various  sizes  of  the  same 
normal  pitch  will  all  work  together  interchangeably,  and  they  are 
probably  the  best  for  change  gears  for  engine  lathes  and  the  like. 
In  the  Rack  and  Pinion  the  wheel  A  may  be  regarded  as  of  in- 
finite diameter  when  the  pitch  line  becomes 
the  straight  line  AC,  Fig.   167.     The  base 
circle  also  becomes  infinite  and  at  an  infinite 
distance  from  C,  so  that  the  line  CD  is  infi- 
nite and  the  tooth  curve  FG  for  the  rack 
becomes   a   straight    line   perpendicular    to 
DE,  the  face  and  flank  being  both  parts  of 
F\      *       one  and  the  same  straight  line.     The  tooth 
FIG.  167.  curve,  HI,  remains  the  same  as  before,  for  a 

like  inclination  of  DE.  The  addendum  line  for  the  rack  should 
never  go  above  E. 

For  the  Annular  Wheel  and  Pinion,  the  tooth  curves  for  the 
former  become  concave  as  at  FG,  Fig.  168, 
while  the  pinion  remains  the  same  as  be- 
fore for  the  same  inclination  of  CED;  and 
for  this  a  group  of  pinions  of  the  same 
pitch  will  interchange  with  this  wheel. 

In  the  present  case  there  can  be  no 
contacts  between  D  and  E,  but  may  ex- 
tend from  E  through  C  to  infinity,  except 
for  interference  at  L. 

In   these   gears,   as  well   as   in  epicy- 


FIG.  168. 


cloidal,  there  maybe  interference  of  teeth,  as  shown  in  an  exaggerate 


CIRCULAR   GEARING. 


145 


ed  view  at  KL  and  KM,  the  tooth  curves  intersecting  at  K.  This  is 
most  likely  to  occur  where  the  pinion  is  relatively  large.  To  examine 
a  case  for  interference,  draw  the  pitch  circles  and  on  them  step  off 
equal  pitch  distances  from  C  to  M  and  N,  and  through  these 
points  draw  the  involute  tooth  curves,  as  shown.  If  these  .curves 
do  not  intersect,  as  at  K,  within  the  addendum  of  the  pinion  and 
the  circle  DL  of  the  wheel,  there  will  be 
no  interference  at  that  position,  and,  if 
the  test  is  applied  at  the  intersection  of 
the  addendum  circle  for  B  with  the  base 
circle  to  AD,  with  no  intersection  of 
tooth  curves,  there  will  be  no  inter- 
ference of  teeth  for  the  wheels.  Modifi- 
cations of  amount  of  interference  are 
affected  by  varying  the  addenda  and  the 
inclination  of  CED. 

Different  systems  and  sizes  of  teeth 
may   le   combined  in   a  single  gear,  as 
illustrated  in  Fig.  169  of  a  pair  of  cir- 
cular gears,  2  to   1,   velocity-ratio  constant,  embracing  both   the 
epicycloidal  and  the  involute  forms  of  teeth. 

III.  CONJUGATE  GEARING. 

This  term  is  applied  to  teeth  where  the  tooth  of  one  wheel  is 

assumed  and  the  mating  tooth  for 
the  other  wheel  found  from  it, 
usually  by  the  practical  operation 
of  templets,  as  foreshadowed  in 
Fig.  108  and  more  definitely  shown 
in  Fig.  122,  which  may  be  followed 
fully  here  with  circular  pitch  lines. 


SOME    PARTICULAR    CASES. 

1st.  Flanks    Parallel    Straight 

Lines. 

In  Fig.  170,  A  and  B  are 
pitch  circles  somewhat  removed 
from  tangency  to  better  show  the 
work.  One  of  the  parallel  flanks 
is  CG.  From  C  step  off  equal 
spaces,  marked  1,  2,  3,  4,  etc.,  on 


E 


146 


PRINCIPLES   OF    MECHANISM. 


the  pitch  line  of  B.  Likewise  from  C'  step  off  the  same  equal  spaces 
on  the  pitch  line  A  as  shown,  C  and  C'  being  a  half-tooth  thick- 
ness from  the  line  of  centers.  Draw  \a,  2b,  3c,  etc.,  perpendicular 
to  CG.  Then  with  la,  from  wheel  B  as  a  radius,  draw  a  circle  arc 
about  the  point  1  on  the  wheel  A\  also,  with  b2  as  a  radius,  draw 
&  circle  at  2  on  A.  Likewise  with  J3,  etc.,  for  as  many  points  as 
required.  Then  draw  an  enveloping  curve  F(1'9  which  will  be 
n  correct  face  curve  for  A,  to  work  on  the  flank  CG  as  mating 
tooth  curves.  Other  pairs  of  mating  face  and  flank  lines  may  be 
-drawn,  and  the  teeth  completed,  as  shown  in  dotted  lines. 

Examining  the  mating  curves  drawn,  it  is  plain,  for  example, 
that  the  radius  from  3  to  where  its  circle  is  tangent  to  the  envelope 
C'F  is  a  normal  to  C'F,  so  that  when  the  points  3  of  pitch  line  A 
.and  B  are  tangent  to  each  other  this  normal  will  coincide  with  the 
normal  3c,  and  c  will  be  the  point  of  contact  and  tangency  between 
the  tooth  lines.  Likewise  for  other  points,  so  that  the  face  and 
flank  will  slide  smoothly  with  each  other  throughout,  while  the 
pitch  lines  mutually  roll,  thus  answering  to  required  conditions  for 
tooth  action. 

2d.  Flanks  Convergent  Straight  Lines. 

In  Fig.  171  draw  the  pitch 
lines  and  the  convergent  flanks 
CG.  Step  the  equal  spaces  1, 2, 
3,  4,  etc.,  draw  the  lines  01,  #2, 
etc.,  normal  to  the  flank  line 
CG,  use  these  latter  as  radii,  and 
draw  circle  arcs  as  before  on  the 
pitch  line  A.  The  envelope  C'F 
will  be  the  proper  tooth  face  for 
the  wheel  A,  to  work  on  on  the 
flank  CG,  and  the  complete 
tooth  can  be  drawn  as  in  Fig. 
170. 

3d.  Flanks  Circle  Arcs. 

Draw  the  pitch  circles  A  and  B,  the  line  of  centers,  and  a  tan- 
gent HI  at  H,  Fig.  172.  From  a  point  /on  HI  draw  the  circle  arc 
CG  for  the  flank  of  a  tooth  for  B.  Through  equidistant  points  1,  2, 
3,  etc.,  draw  radii  from  I.  Then  al,  62,  etc.,  are  the  radii  to  use 
,at  points  1,  2,  3,  etc.,  on  A  as  centers  of  arcs,  tangent  to  which 


FIG.  171. 


CIRCULAR   GEARING. 


147 


draw   the   envelope   C'F  for   a   face  of  a  tooth  of  A  that  will 
work  correctly  on  the  flank  CG. 

As  another  example,  the  cen- 
ter 1  may  be  taken  anywhere  be- 
low the  line  HI,  but  no  higher 
above  than  to  be  on  a  tangent 
to  the  pitch  circle  Bat  the  point 
C,  In  which  case  CG  will  be  nor- 
mal to  the  pitch  line.  No  example 
is  to  be  found  of  a  correct  flank 
line  which  is  undercut  as  by 
placing  /  still  higher. 

Again,  in  another  example, 
the  line  CG  may  be  a  parabola, 
•ellipse,  or  other  curve  to  suit  the  fancy,  always  being  so  drawn  as 
not  to  undercut  at  C,  and  in  every  case  the  lines  «1,  62,  etc., 
being  drawn  normal  to  the  curve  CG. 

The  inverse  of  these  cases  is  practicable  where  the  face  of  a  tooth 
of  A  is  assumed  to  be  a  circle,  parabola,  or  other  curve,  and  the 
flank  found  to  match  it  in  a  similar  way. 


CHAPTER  XIII. 
PRACTICAL  CONSIDERATIONS. 

So  far  in  circular  gearing,  we  have  dwelt  upon  the  theory  of 
tooth  curves  which  are  perfect  in  action.  But  the  demands  of 
practice  are  not  always  so  severe  as  to  require  the  teeth  to  be  thus 
accurate.  There  are  other  considerations  also  to  be  noted,  such  as- 
proper  addenda,  clearance,  fillets,  obliquity  of  action,  overlap  of 
action,  practical  methods,  approximate  teeth,  etc. 


ADDENDA   AND    CLEARANCE. 

At  Fig.  110  is  given  a  set  of  rules  for  proportioning  the  teeth, 
which  are  among  the  simplest  of  quite  a  variety.  In  cast  gearing 
more  clearance  is  required  than  in  machine-dressed  teeth,  and  more  as 
cast  at  some  foundries  than  at  others,  according  to  care  in  moulding. 

The  rule  for  circumferential 
pitch  at  Fig.  110  will  answer  for 
the  average  of  care  in  moulding, 
and  for  ordinary  sizes.  For 
very  large  teeth  the  propor- 
tion for  clearance  is  too  great, 
while  for  very  small  cast  teeth 
it  will  often  be  found  too  small, 
so  that  the  best  rule  will  probably  be  the  one  stated  varied  to  suit 
the  judgment  of  the  draftsman  in  each  particular  case. 

The  rule  as  to  diametral  pitch  is  best  adapted  for  cut  or  machine 
dressed  gear  teeth,  where  a  comparatively  slight  bottom  clearance 
is  sufficient  and  a  much  less  side  clearance.     The  Brown  &  Sharps 
cutters  and  instructions  are  apropos  here. 

148 


-  173. 


PRACTICAL   CONSIDERATIONS. 


149 


In  Fig.  173,  for  cast  teeth,  cd  is  the  pitch  circle,  e  the  addendum 
circle,  /  the  dedendum  or  root  circle;  c  being  equal  to  5/11,  d 
equal  to  6/11,  a  equal  to  3/10,  and  b  equal  to  4/10  the  pitch,  ac- 
cording to  the  rule  of  Prof.  Willis,  as  given  at  Fig.  110. 

To  Strengthen  the  Teeth,  root  fillets  are  generally  struck  in  as  at 
jk,  which  only  extend  to  a  part  of  the  bottom  surface  at  k  and  up  to 
.a  comparatively  small  portion  of  the  flank  at/.  In  most  cases  the 
bottom  of  the  space  may  be  limited  by  a  circle  arc,  ghi,  tangent  to 
the  flank  lines  at  g  and  i,  and  to  the  root  circle,  thus  simplifying 
the  shape  of  the  bottom  of  the  space  and  greatly  strengthening  the 
teeth.  To  test  this  matter  the  so-called  "  clearing  curve"  may  be 
drawn — a  curve  that  would  be  described  by  the  corner  of  the  point 
of  the  tooth,  as  in  Fig.  174. 

To  Find  the  Possible  Clearing  Curve  described  by  the  tooth 
corner  b,  Fig.  174,  as  it  enters  the  space  def,  the  side  ab  acting  against 
the  side  de,  make  equal  spaces  1,  2,  3,  etc., 
then  with  bi  for  a  radius  describe  a  circle 
arc  from  the  point  1  near  d',  again,  with  b2 
for  a  radius  describe  the  arc  from  2  in  the 
space  near  d,  etc.,  for  as  many  points  and 
arcs  as  required.  Then  the  enveloping 
curve  to  these  arcs,  as  shown,  is  the  possi- 
ble clearing  curve.  This  curve,  of  course, 
extends  only  the  addendum  distance  be- 
low the  pitch  line.  Now  a  circle  arc, 
drawn  in  as  ghi,  Fig.  173,  where  a  =  3/10 
and  b  =  4/10,  will  usually,  if  not  always,  FIG.  174. 

be  found  to  go  outside  of  this  possible  clearing  curve,  and,  as  it  is 
the  simplest  form  of  outline  for  the  bottom  of  the  space  as  well  as 
gives  greater  strength  of  tooth  than  does  the  smaller  fillets  jk, 
Fig.  173,  it  is  recommended.  Indeed,  if  ever  found  necessary,  it  is 
usually  advisable  to  go  a  little  deeper  to  get  in  the  circle  arc  rather 
than  adopt  two  smaller  arcs.  The  circle  arc  is  dotted  in,  and  is  seen 
to  go  considerably  below  the  curve  enveloping  the  circles  struck 
from  2,  3,  etc.,  Fig.  174. 


THE   PATH   OF   CONTACT. 


The  Path  of  Contact  between  a  pair  of  operating  teeth  of  mated 
gear  wheels  is  the  line  followed  by  the  touching  point  between  the 
pair  of  teeth,  and  differs  in  form  in  different  systems  of  gearing. 


150 


PRINCIPLES   OF   MECHANISM. 


FIG.  175. 


In  Epicycloidal  Teeth  it  is  the  Describing  Circle  Itself,  in  a  cen- 
tral position,  or  that  of  common  tangency 
with  both  pitch  circles,  as  at  C,  Fig.  175r 
where  CF  is  the  describing  circle.  This  may 
be  shown  by  supposing  CD  equal  to  CE,  and 
that  the  describing  circle  has  rolled  from 
where  the  tracing  point  F  was  at  D  over  to 
the  position  shown,  and  likewise  from  E 
over  to  the  same  position.  The  rolled  curves 
DF  and  EF  will  be  tangent  to  each  other  at 
F9  because  CF  is  a  normal  to  both  curves,  since  C  is  an  instanta- 
neous axis  of  motion  of  the  tracing  point.  Therefore  F,  the  point  of 
contact  of  the  tooth  curves  DF  and  EF  is  on  the  describing  circle. 
Hence  the  point  of  contact  F  follows  along  on  the  describing  circle 
CF  from  C  to  F,  the  tooth  curves  engaging  at  C  and  move  toward 
the  position  shown,  making  the  describing  circle  CF  the  path  of 
contact. 

In  Involute  Gearing,  it  is  plainly  seen  from  Fig.  166  that  the 
path  of  contact  is  on  the  line  DC'E. 

In  Other  Cases,  the  Path  of  Contact  may  Readily  be  Found.  To 
illustrate,  let  us  find  it  for  the  teeth 
of  Fig.  172  as  shown  in  Fig.  176. 
Draw  radii  from  B  to  the  points  1,  2, 
3,  where  normals  to  the  flank  curve 
Cc  cut  the  pitch  circle.  Draw  circle 
arcs  through  the  points  a,  b,  c,  where 
the  normals  meet  the  flank  curve,  and 
extend  them  to  meet  the  radii  from  B. 
Then,  evidently,  the  flank  curve  is 
in  contact  at  «,  with  its  mating  face, 
when  the  point  1  is  at  H  on  the  line  of  centers,  and  a  at  i,  where 
im  =  ae;  thus  making  i  one  point  in  the  path  of  contact.  Like- 
wise, jn  =  bff  ko  =  eg,  etc. ;  thus  giving  the  curve  Hijk  as  the 
path  of  contact  for  the  pair  of  tooth  curves  of  Fig.  172  as  above 
the  pitch  lines.  Usually  the  path  extends  below  the  point  of  com- 
mon tangency  of  the  pitch  lines  toward  A,  for  the  mating  tooth 
curves  below  the  pitch  lines  as  well  as  above  toward  B,  and  this 
path  is  always  a  curve,  except  in  involute  teeth. 

By  the  Inverse  of  this  the  Path  of  Contact  may  be  Assumed,  and 
the  tooth  curves,  face  and  flank,  may  be  found  therefrom,  as  was. 


FIG.  176. 


PRACTICAL   CONSIDERATIONS.  151 

done  by  Prof.  Edward  Sang,  and  explained  in  his  remarkable  treat- 
ment of  the  subject  of  gear  teeth  published  some  years  ago. 

The  Path  of  Contact  is  Limited  in  Practice  where  the  addendum 
circles  cut  the  path  lines.  Thus,  iii  Fig. 
177,  draw  the  addendum  circles  EF  and 
DF.  Then  the  contacts  for  the  epicycloi- 
dal  teeth  of  the  wheels  A  and  B  must 
begin  and  end  on  the  S-shaped  line  or 
limited  path  DCE.  One  pair  of  teeth 
should  engage  at  D  as  soon  as  the  preced- 
ing pair  quit  contact  at  E,  but  usually  it 
is  sooner.  In  involute  teeth,t  the  contact 
cannot  possibly  extend  beyond  the  points  FIG.  177. 

D  and  E,  Fig.  166,  so  that  the  addendum  circles  should  not  over- 
reach those  points  as  has  been  stated. 

The  Acting  Part  of  the  Flank  of  a  tooth,  that  is,  the  entire 
portion  that  engages  with  its  mating  face,  is  simply  that  which  ex- 
tends from  the  point  D  to  the  pitch  line  of  B,  or  from  E  to  the  pitch 
circle  of  A,  Fig.  177.  This  portion  of  the  flank  is  much  shorter,, 
usually,  than  its  mating  face,  thus  giving  cause  for  more  rapid 
wear  on  the  flank  than  face,  from  which  it  appears  that  gear  teeth 
will  wear  out  of  shape  and  not  into  it,  so  that  if  teeth  are  ever 
correct  in  form,  they  must  be  made  so. 

The  Line  of  Action  of  the  Pressures  of  Contact  between  the 
teeth,  neglecting  friction,  is  always  in  the  line  of  the  normal,  as 
CF,  Fig.  175;  this  line  being  therefore  properly  called  the  line  of 
action.  In  involute  gearing  it  is  always  one  and  the  same  straight 
line,  DCE,  Fig.  166,  but  in  all  other  forms  of  teeth  it  is  a  line  of 
varying  inclination,  like  a  straight  line,  iHt  jlf,  kH,  etc.,  Fig.  176- 

The  tendency  of  the  teeth  to  crowd  the  wheels  apart  depends 
upon  an  intermediate  obliquity  of  this  line  of  action,  the  same 
being  reduced  as  the  maximum  obliquity  is  reduced  for  a  given  sys- 
tem of  teeth. 

Thus  in  epioycloidal  gears,  the  larger  the  describing  circles,  CD 
and  CE,  Fig.  177,  the  less  the  obliquity  and  the  tendency  to  crowd. 
The  maximum  obliquity  of  the  line  of  action  with  the  common1 
tangent  to  the  pitch  lines  is  that  of  the  straight  line  through  /> 
and  C  on  the  one  side,  and  the  line  EC  on  the  other. 


152 


PEIKCIPLES   OF   MECHANISM. 


The  Blocking  Tendency. 

The  tendency  of  a  pair  of  teeth  to  "  block  "  or  stop  the  wheels 
is  greater  in  approaching  action  than  receding,  and  increases  with 
the  obliquity  of  action;  so  that  in  wheels  of  few  teeth  the  maximum 
obliquity  of  the  line  of  action  should  be  kept  as  small  as  possible, 
as  already  explained  under  non-circular  gearing,  and  sometimes  it 
may  be  advisable  to  make  the  addenda  unequal  to  favor  the  obli- 
quity in  approaching  action. 

Unsymmetrical  Teeth. 

These  may  be  made  when  desired,  by  using  smaller  describing 
circles  for  one  side  of  a  tooth  than  for  the  other,  for  epicycloidal 
teeth,  or  greater  obliquity  of  the  line  of  action  in  involute,  etc. 
Pig.  169  illustrates  several  such  teeth. 

To  Draw  the  Tooth  Curves  in  Practice. 

Epicycloidal  Tooth  Curves  may  be  drawn  as  in  Fig.  178,  where 
QB         /  A  and  B  represent  a  pair  of  pitch 

circles  for  which  it  is  desired  to 
draw  a  pair  of  tooth  curves  of  this 
form  for  teeth  with  concave  flanks, 
the  work  all  being  completed  on 
the  drawing-board  and  with  only 
the  ordinary  drawing  instruments. 
Having  the  pitch  circles  drawn  at 
a  distance  apart,  if  need  be,  for  per- 
spicuity, adopt  some  diameter  of 
describing  circle  CH  =  C'H,  such 
as  has  a  diameter  less  than  the 
radius  O 'B  as  for  concave  flanks 
for  B.  Then,  with  the  dividers,  ' 
describe  the  circle  in  several  posi- 
tions on  A,  and  in  B,  tangent  to 
the  pitch  circles  as  shown.  Then 
with  C  as  the  origin  of  a  tooth  curve,  step  off  with  the  spacing  di- 
viders on  A,  as  at  abc,  etc.,  to  near  the  point  of  tangency  with  A  of 
some  drawn  circle  F9  say  the  first  one,  and  then,  without  lifting 
the  dividers  from  the  paper,  step  backwards  on  this  circle  an  equal 
number  of  steps  to  F,  and  note  the  exact  point.  Then  beginning 
At  O  again,  step  along  on  A  to  near  the  tangency  with  the  next 


PRACTICAL   CONSIDERATIONS. 


153 


circle,  and  then  back  on  that  circle  an  equal  number  of  steps  to  G, 
.and  carefully  note  that  point.  So  proceed  until  the  desired  num- 
ber and  frequency  of  noted  points  is  obtained.  Then  the  epi- 
cycloid, CFG,  may  be  drawn  in  through  the  points  noted  by  aid  of 
an  irregular  curve. 

In  stepping  from  C  along  on  A  it  is  not  necessary  to  come  out 
exactly  at  the  point  of  tangency  with  the  circle  F,  G,  etc.,  but  the 
nearest  half-step  or  less  from  it,  and  then  turn  back  on  that  circle 
to  the  point  F  or  G,  as  the  case  may  be.  Neither  is  it  necessary,  as 
F  is  noted,  to  go  back  to  C,  but  without  lifting  the  dividers  we  may 
retrace  the  steps  from  F  to  A  and  thence  along  A  to  near  the  next 
point  of  tangency  and  then  backwards  on  that  next  circle,  to  G,  etc. 

Proceed  likewise  for  all  the  tooth  curves  to  be  drawn,  as,  for  in- 
stance, the  mating  flank  for  C,  F,  G,  using  the  same  describing 
circle,  resulting  in  the  points  C",  Z>,  E,  etc.,  in  B.  Also  a  face  for  B 
in  the  points  L,  M,  N,  etc.,  and  its  mating  flank  /,  J,  K  in  A,  using 
the  same  describing  circle  for  both.  We  then  have  a  complete  tooth 
profile,  GFC1J,  for  A,  and  another,  NL CDE,  for  B.  Templets  may 
be  formed  to  these  curves  and  used  to  finish  drawing  all  the  teeth 
of  the  gear  wheel  A  or  B,  either  on  the  drawing-board  or  on  the 


FIG.  179. 

wooden  pattern  being  made  for  use  in  casting.  When  the  flank  is 
ready  for  it  prepare  a  templet  of  cardboard,  thin  piece  of  wood,  or 
zinc  plate.  The  latter  may  be  blackened  with  a  chemical  as  an  ex- 
cellent preparation  for  visibility  of  lines  drawn  upon  it.  Lay  off 
the  curve  GCK,  for  instance,  with  accuracy.  Also,  if  desired,  the 
addendum  circle  at  P,  Fig.  179,  and  the  root  circle  at  0,  to  fit  the 
root  curve  of  Fig.  173.  At  R  cut  an  opening  with  a  sharp  point  R 
coming  exactly  to  one  of  the  pitch  points  T,  R,  V,  so  that  CR 


154  PRINCIPLES   OF   MECHANISM. 

equals  a  half-tooth  thickness.  Also  at  S  cut  the  templet  to  a  point 
that  shall  just  fit  upon  the  pitch  circle.  This  gives  us  the  required 
tooth  templet.  Then,  when  this  templet  OCPRSis  placed  at  any 
pitch  point,  as  R,  the  edge  of  the  templet  PCO  is  in  just  the  posi- 
tion for  drawing  a  scriber  along  the  edge  of  the  templet  to  trace 
the  whole  tooth  profile,  including  addendum  and  root  curve.  Then 
the  templet  is  to  be  shifted  along  the  pitch  line  till  the  point  of  the 
templet  at  R  just  coincides  with  the  next  pitch  point  T,  and  the 
point  S  with  the  pitch  line,  when  the  next  tooth  profile  may  be 
traced,  as  dotted  at  QU.  Proceed  likewise  around  the  wheel-blank, 
or  the  drawing  on  the  drawing-board.  Then  the  templet,  being 
made  of  thin  material,  may  be  turned  over  and  the  other  sides  of 
the  teeth  all  drawn,  and  the  outlines  of  the  teeth  completed,  as, 
shown  for  the  one  OQU. 

A  Radius  Rod,  if  preferred,  may  be  used  instead  of  the  points 
R  and  S,  the  templet  being  mounted  as  at  X  on  the  rod  AX\)j 
tacks  or  screws,  and  a  center  pin  struck  through  the  radius  rod,  and 
at  the  exact  center  point  A,  by  means  of  which  the  templet  may  be 
swung  around  the  wheel  and  always  remaining  in  the  exact  tooth- 
curve  position ;  it  being  only  necessary  to  stop  and  hold  it  at  each  of 
the  several  pitch  points,  while  tracing  the  tooth  curve  for  the  same. 
The  templet  may  be  turned  over  on  the  radius  rod  and  the  other 
sides  of  the  teeth  drawn. 

For  Involute  Teeth,  as  in  Fig.  180,  let  A  and  B  represent  a  pair 
of  pitch  lines  for  which  it  is  desired  to  draw  this  form  of  tooth. 
Draw  a  line  DCE  through  the  pitch  point  C  and  at  an  angle  of 
about  75  degrees  with  the  line  of  centers  AB.  Also  draw  AD  and 
BE  from  the  centers  A  and  B  perpendicular  to  DE,  and  circles- 
tangent  to  DE  as  shown.  To  the  last-named  circles  draw  a  series 
of  tangents  JL,  GM,  KN,  10,  as  many  as  required.  Then  with 
spacing  dividers  step  from  C  along  on  CD  to  near  the  tangency  D+ 
then  backward  on  the  circle  an  equal  number  of  steps  to  F,  and  note 
that  point.  Without  lifting  the  dividers,  step  back  past  D  to  near 
L  and  out  on  LJ  to  J  an  equal  number  of  steps  and  note  the  point 
J.  Then  back  on  JL  and  on  to  near  M  and  out  on  MG  to  G  and  note 
that  point.  So  proceed  for  as  many  lines,  JL,  GM,  etc.,  as  the  de- 
sired frequency  of  the  points  F,  C,  J,  G  calls  for.  Then  trace  the 
curve  through  these  points,  which  curve  will  be  an  involute  to  the- 
circle  CDM,  exactly  in  theory,  and  very  nearly  so  by  this  process  of 
drawing,  and  to  serve  for  a  tooth  curve  for  wheel  A.  Likewise  the 
involute  HCKI  is  to  be  drawn  to  serve  for  a  tooth  curve  for  B» 


PRACTICAL  CONSIDERATIONS. 


155- 


Then  the  addendum  circles  and  the  root  curves  are  to  be  drawn  in,, 
giving  the  full  tooth  profiles,  corresponding  to  PCO  of  Fig.  179,. 
when  templets  may  be  made  and  applied  as  explained  at  Fig.  179. 
The  line  DOE  was  said  to  be  drawn  at  about  75  degrees  with  AB, 
but  this  angle  is  arbitrary,  some  preferring  it  at  more  and  some  less 
than  75  degrees.  Probably  the  best  criterion  to  follow  as  to  this. 


FIG.  180. 

angle  is  the  judgment  of  the  designer  as  to  form  of  teeth  a  partic- 
ular angle  gives,  except  that  the  addendum  circle  for  A  should  not 
reach  beyond  the  point  E,  nor  the  addendum  circle  for  B  reach 
beyond  Z>,  on  the  penalty  of  interference  of  teeth;  and,  at  the  same 
time,  the  normal  pitch  of  the  teeth  should  not  exceed  the  portion 
of  DE  as  intercepted  between  the  addendum  circles.  The  normal 
pitch  is  the  distance  from  one  tooth  to  the  corresponding  point  on 
the  next,  as  measured  along  on  the  line  DE,  this  line  being  normal 
to  all  tooth  curves  that  intersect  it  between  D  and  E. 

Approximate  Teeth  in  Practice. 

In  many  cases,  a  simple  circle  arc  will  answer  the  requirement 
for  a  face  curve  and  also  flank,  especially  in  gear  patterns  where  the 
inaccuracy  of  moulding  of  the  gears  in  moulding  sand  enters  the 
account;  and  for  moderate  sizes.  But  it  is  easy  to  select  a  curve 
that  will  approximate  the  epicycloid  or  the  involute  closer  than 
the  circle  will,  as,  for  instance,  an  ellipse,  though  a  series  of  ellipses 
of  various  sizes  would  be  needed  to  select  from  in  a  particular  case 
for  the  best  results.  To  avoid  this,  a  parabola  or  hyperbola  may,  in 
a  single  curve,  meet  all  sizes  of  teeth,  as  well  as  a  series  of  ellipses, 


156 


PRINCIPLES   OF   MECHANISM. 


II 


FIG.  181, 


rbut  how  to  locate  the  curve  to  best  approximate  the  true  tooth 
without  drawing  it  would  be  a  serious  question  with  a  parabola  or 
hyperbola.  Further  thought  suggests  a  spiral  as  having  advan- 
tages, and,  finally,  the  logarithmic  spiral,  by  reason  of  its  peculiar 
and  simple  properties,  is  perceived  to  be  the  best  adapted  of  all 
curves  for  this  purpose.  Thus  it  is  well  known  that  a  normal  to 
this  spiral  at  any  point,  at  will  be  tangent  to  an  equal  logarithmic 

spiral    EcF,     Fig.     181; 

rand  that  ac  will  equal 
the  length  of  the  curve 
cE,  and  also  equal  the 
radius  of  curvature  of  the 
logarithmic  spiral  Ea  G 
at  a.  This  shows  that 
the  two  equal  copolar 
logarithmic  spirals  EaG 
and  EcF  are  mutual  in- 
volute and  evolute  curves. 
Also,  it  is  known  that 
the  triangle  Eac  is  right- 
angled  at  E. 

It  is  found  by  trial  on  a  carefully  made  drawing  that,  taking  any 
point  d  as  a  center  on  ac  produced,  and  drawing  a  circle  through 
n  as  dotted,  the  circle  will  cut  the  spiral  at  such  point,  b,  that  the 
length  cdy  divided  by  the  length  be,  will  be  very  nearly  a  constant, 
at  least  when  the  obliquity  of  the  spiral  is  such  that  EC  equals  2Ea, 
as  in  the  case  of  the  logarithmic  spiral,  which  is  found  to  best  fit 
the  epicycloidal  tooth  curve. 

Now  draw  a  pitch  line  IJH  for  a  gear  wheel  and  tangent  to  ad 
at  J9  so  that  tfj^will  equal  the  half-tooth  thickness.  Then  the 
-epicycloidal  tooth  curve,  suitable  for  a  tooth,  so  drawn  as  to  pass 
through  #,  will  also  pass  very  nearly  through  Z»,  when  the  latter  is 
taken  at  the  addendum  circle;  due  regard  being  paid  to  the  pitch 
diameters  and  tooth  numbers  in  drawing  the  epicycloid. 

The  Template  Odontograph  has  been  formed  of  this  spiral  with 
EC  =  2Ea,  or  with  the  tangent  of  obliquity  equal  to  %,  and  is 
therefore  a  curve  which  very  closely  approximates  the  epicycloidal 
tooth  curve.  It  forms  at  once  a  universal  and  ready-made  tooth 
templet  for  the  draftsman,  which  has  been  generally  accepted  as 
being  a  good  substitute  for  the  true  tooth  curve.  The  instrument 
is  accompanied  by  tables  for  use  in  setting  it  on  the  pitch  line  in 


PRACTICAL   CONSIDERATIONS. 


157 


the  right  position  for  drawing  the  various  kinds  of  teeth,  such  as; 
flanks  radial,  flanks  concave,  interchangeable  sets,  involute  teeth,, 
etc.* 

The  instrument,  full  size,  is 
about  four  by  six  inches  and  is 
shown  in  Fig.  182.  The  curved 
edge  at  A  is  graduated  and  num- 
bered for  a  length  of  3  inches.  To 
set  the  instrument  in  position, 
"  settings "  are  made  out  from 
accompanying  tables,  as,  for  ex- 


FIG.  182. — Odontograph  set  at 
setting  number  2.50  on  line 
AC. 


ample,  2.50.  A  line  ABC  is 
drawn,  when  the  2.50  is  brought 
to  the  line  at  A,  while  the  curved 
edge  at  B  is  brought  just  tangent  to  the  same  line  ABC. 

To  set  the  instrument  for  drawing  the  tooth  of  Fig.  183,  draw 
the  tangent  AHCE  to  the  pitch  line  at  C,  the  middle  point  of  the 

tooth,  also  a  tangent  BD  at 
H,  the  side  of  a  tooth.  Then, 
with  "  settings  "  made  out 
from  the  tables,  place  the 
instrument  on  the  tangent 
HCE  as  explained,  and, 
while  retaining  it  there, 
draw  the  scriber  along  the 
FIG.  183. — How  to  place  the  Odontograph.  edge  from  D  outward,  giving- 
the  face  curve.  Similarly  with  the  instrument  set  on  BD  as  shown, 
draw  the  scriber  from  D  inward,  giving  the  flank  curve.  The 
addendum  circle  and  root  curves  may  then  be  drawn  in  as  pre- 
viously explained,  when  the  full  tooth  profile  is  completed. 

This  Templet  Odontograph,  by  means  of  screw-holes  shown,, 
may  be  mounted  on  a  radius  rod  and  swung  around  a  center  pin, 
for  marking  all  the  tooth  curves,  as  explained  at  Fig.  179.  We 
thus  have,  in  this  instrument,  a  convenient  ready-made  tooth  tem- 
plet for  all  cases. 

The  accuracy  of  results,  as  compared  with  those  of  other  ap- 
proximate methods,  is  discussed,  and  examples  are  given  in  the* 

*  For  a  full  description  of  the  instrument,  formulas  for  calculating  "set- 
tings," tables  of  settings,  etc.,  see  Van  Nostrand's  Science  Series,  No.  24,  and 
Van  Nostrand's  Engineering  Magazine  for  1876  ;  also  a  pamphle  accompany- 
ing the  instrument,  all  found  in  the  instrument  stores. 


158  PRINCIPLES   OF   MECHANISM. 

articles  cited  in  reference.  Involute  teeth,  as  well  as  the  various 
cases  of  rack-and-pinion  annular  wheels,  etc.,  are  drawn  by  the 
instrument. 

The  Willis  Odontograph  is  a  well-known  instrument  for  locat- 
ing the  centers  of  circle  arc  tooth  curves  in  such  a  way  that  p,  face 
is  of  one  circle-arc  and  the  flank  of  another,  for  teeth  approx- 
imating those  of  the  epicycloidal  form  in  interchangeable  sets, 
where  a  wheel  of  12  teeth  has  radial  flanks.  This  instrument  is 
of  special  interest,  as  being  the  first  odontograph:  originated  some 
.sixty  years  ago  by  Prof.  Eobert  "Willis. 

The  instrument  is  founded  on  principles  made  clear  by  Fig. 
184,  where  A  C  and  BC  are  the  radii  of  the 
pitch  circles,  CDH  the  generating  circle 
to  roll  on  A,  to  describe  a  face  of  a  tooth 
of  A  and  to  roll  inside  of  B  to  describe 
the  flank  for  B,  that  is  to  mate  with  the 
face  of  A.  Take  the  center  of  CDH  on 
the  line  AB. 

Assume  a  point  D  on  the  describing 
circle  CDH,  and  draw  DCF,  DH,  CI 
parallel  to  DH,  BIF,  and  AEL  Then  01 
and  DH  will  be  perpendicular  to  DCF, 
because  CDH  is  in  a  semicircle;  and  E 
•will  be  the  center  of  curvature  at  D  of  an  epicycloid  drawn  through 
D  by  a  tracer  at  D  in  the  circle  CDH  used  as  the  describing  circle, 
as  the  latter  is  rolled  on  the  pitch  circle  A.  Also,  ^will  be  the 
center  of  curvature  at  D  of  a  hypocycloid  drawn  through  D  by  a 
tracer  at  D,  in  the  describing  circle,  as  the  latter  rolls  inside  tho 
pitch  circle  B* 

*A  proof  of  this  is  given  in  Van  Nostrand's  Engineering  Magazine  for 
1878,  Vol.  XIX,  page  313,  viz.: 

CD  =  HI  :  CE  :  :  AH  :  AC. 

_    CD  X  AC  _  CD    X  AC  _„• 

..  CE-  AH       ,   and  DE  -  —^- 

which  is  the  expression  for  the  radius  of  curvature  at  D  for  an  epicycloid 
passing  through  D  as  generated  by  rolling  the  describing  circle  CDH  on  the 
pitch  line  A. 

In  a  similar  way,  the  radius  of  curvature  at  D  of  the  hypocycloid  passing 
through  D,  as  generated  by  rolling  the  describing  circle  CDH  on  the  pitch 
line  B,  is 

SC\ 

f  BE)' 

See  Da  vies  and  Peck's  Mathematical  Dictionary,  page  222,  for  formulas  for 
radii  of  curvature. 


PKACT1CAL   CONSIDERATIONS.  159 

Now,  as  E  and  F  are  centers  of  curvature  of  the  mating  epi- 
cycloid and  hypocycloid  tooth  curves,  it  is  plain  that,  by  taking  the 
points  E  and  F  as  centers  and  describing  circle  arcs  through  D, 
these  arcs  will  approximate  the  epicycloidal  curves  and  serve  as 
mating  tooth  curves  with  corresponding  approximation  to  accuracy. 

Without  reference  to  analysis,  regard  IGF  as  a  triangular  tem- 
plet, swinging  about  a  pin  at  /.  For  a  movement  corresponding  to 
that  of  the  action  of  face  and  flank,  the  edge  CF  will  not  depart 
far  from  intersection  of  AB  at  C.  This  edge  line,  CF,  would  have 
nearly  the  same  movement  if,  instead  of  one  side  of  a  triangle,  it 
were  jointed  at  E  to  a,  rod  AE,  since  E  is  on  the  straight  line  AI9 
and  would,  in  either  case,  move  in  a  circle  arc  normal  to  A  I.  Also, 
for  a  like  limited  movement,  if  F  of  the  line  CF  were  jointed  to  a 
rod  BF,  the  point  F  would  move  in  nearly  the  same  path,  so  that 
motion  transmitted  from  A  to  B  through  the  medium  of  rods  AE, 
EF,  and  FB,  jointed  at  E  and  B,  would  maintain  nearly  constant 
velocity-ratio  for  the  limited  movement  considered,  as  required  for 
the  tooth  curves. 

If  now,  with  E  and  Fas  centers,  circles  be  struck  through  D 
for  a  pair  of  tooth  curves  for  gears  A  and  B,  their  velocity-ratio 
would  be  the  same  as  that  for  the  rods  connecting  the  axes  A  and 
B,  because  the  distance  between  the  centers  E  and  F  would  re- 
main constant  in  either  case.  This  last  consideration  holds  for 
whatever  point  near  D  in  the  circle  CDH  selected,  as  that  through 
which  the  circle-arc  tooth  curves  be  struck. 

Hence,  to  draw  a  pair  of  approximately  correct  tooth  curves  as 
circle  arcs,  for  the  pitch  lines  A  and  B,  Fig.  184,  draw  the  lines 
AB,  CF  produced,  CI  perpendicular  to  CF,  AI,  and  BIF.  Then, 
with  E  and  F  as  centers,  draw  a  pair  of  circle  arcs  through  some 
point  on  CF  produced.  Thus,  Fig.  184  is  general,  but  for  con- 
venience of  millwrights  Prof.  Willis  limits  the  diagram,  the  point 
through  which  the  teeth  are  drawn  being  taken  a  half-pitch  from 
C  toward  D,  and  may  sometimes  be  the  point  D,  as  when  the  line 
CF  is  drawn,  as  Prof.  Willis  takes  it,  at  an  angle  of  75  degrees  with 
AB. 

To  obtain  a  convenient  odontograph,  Prof.  Willis  assumes  the 
points  /  and  D  to  be  on  a  circle  CDHI  of  constant  diameter  for  a 
given  pitch  and  to  equal  the  radius  of  a  wheel  of  twelve  teeth,  for 
which  ,D  will  be  a  half-pitch  from  C,  and  the  line  DCFat  an  angle 
of  75  degrees.  This  makes  all  the  tooth-circle  arcs  pass  through 
the  point  D  when  DE,  the  radius  of  a  face,  will  be  the  radius  of 


160  PRINCIPLES   OF   MECHANISM. 

curvature  at  D  of  an  epicycloid  through  D  generated  by  rolling  the 
circle  CDH on  the  pitch  circle  A',  and  DFthe  radius  of  curvature 
of  the  hypocycloid  in  B  through  D,  thus  giving  for  the  tooth  curves 
the  osculating  circles  of  the  epicycloid  and  hypocycloid  at  D.  A 
carefully-made  drawing  will  show  that  the  point  D  for  the  above 
limit  will  be  a  little  above  the  half-height  of  the  tooth  face;  while 
it  is  probable  that  a  preferable  result  is  obtained  if  it  be  a  little  be- 
low, as  for  the  case  where  a  15-toothed  wheel  has  radial  flanks  with 
CD  a  half-pitch  and  the  line  CF  at  an  angle  of  78  degrees  with  AB. 
The  Willis  Odontograph,  shown  at  agF,  Fig.  185,  locates  the 


centers  E  and  F  of  Fig.  184  with  only  the  brief  diagram  of  Fig. 
185,  where  CO'  is  one  pitch  in  the  pitch  circle,  D  the  middle  or  half 
pitch  point,  AC  and  AC'  radii,  and  DL  and  DK  a  face  and  flank 
curve  or  circle  arc  respectively,  drawn  to  the  centers  E  and  F. 
These  centers  are  found  by  placing  the  odontograph  at  C'E  and  at 
CF,  with  the  leg  Ca  on  the  radius,  and  noting  the  points  E  and  F 
by  aid  of  a  table  from  which  values,  or  distances  C'E  and  CF  are 
taken,  according  to  radius  of  wheel  being  drawn,  and  noted  in  the 
scale  gCF.  When  the  tooth  profile  KDL  is  thus  drawn,  a  profile 
templet  may  be  formed  as  explained  at  Fig.  179,  and  the  tooth  out- 
lines drawn  for  the  complete  wheel. 

This  odontograph  is  made  in  metal  or  cardboard,  the  latter  being 
larger  and  having  the  necessary  tables  printed  thereon. 

A  simple  odontograph  was  brought  out  by  Prof.  Willis,  giving 
centers  of  tooth-circle  arcs,  when  the  whole  profile  is  formed  of 
one  arc,  thus  approximating  involute  teeth.  This  instrument  sim- 
ply gives  the  centers  D  and  E,  Fig.  166,  and  is  of  so  little  help  in 
finding  D  and  E  as  to  be  in  slight  demand. 

The  Three-Point  Odontograph  of  Geo.  B.  Grant  is  so  called  be- 


PRACTICAL   CONSIDERATIONS.  161 

cause  its  application  results  in  a  circle  arc  struck  through  three  points 
Cab,  Fig.  186,  in  the  actual  epicycloidal  face  curve  CD;  one  point, 
C,  being  at  the  pitch  line  ICH,  a  second  at  the  middle  of  the  face 
at  a,  and  the  third  at  b,  where  the  addendum  circle  intersects  the 


bx''D 


epicycloid.  The  position  of  the  center  E,  of  a  circle  which  will  pass 
through  the  three  points  C,  a,  b,  is  calculated,  and  the  radius,  EJ, 
"called  "  Had./''  and  distance,  HE,  called  "  Dis.,"  are  tabulated  for  a 
large  variety  of  pitches  and  sizes  of  gears. 

The  flank  is  treated  in  like  manner  for  three  points  (7,  d,  e  with 
the  radius  Fd  and  distance  Fl  determined  for  various  sizes  and 
tabulated. 

Tables  for  unit  pitch  may  thus  be  made  out  for  all  varieties  of 
teeth,  not  only  of  the  epicycloidal  order,  but  involute,  conjugate,  etc. 

The  application,  in  addition  to  drawing  the  pitch  circle  and 
pitch  point,  requires  simply  that  "  Dis."  be  taken  from  the  table, 
multiplied  by  the  pitch  and  laid  off  on  a  radius  from  the  pitch  line, 
as  HE,  and  a  circle  drawn  through  the  point  concentric  with  the 
pitch  circle.  Then  the  "Rad.,"  taken  from  the  table,  multiplied 
by  the  pitch,  is  to  be  taken  iu  the  dividers,  when  circular  faces  Cab 
may  be  struck  through  all  the  pitch  points  C  from  centers  taken  on 
the  circle  through  E. 

In  like  manner,  the  circular  flanks  are  to  be  struck  in  from  cen- 
ters on  the  circle  through  F  by  aid  of  values  for  "  Rad."  and  "  Dis." 
taken  from  the  table. 

The  maximum  error,  or  deviation  of  these  circle-arc  faces  from 
the  true  epicycloid,  between  (7  and  b,  is  stated  to  be  less  than  the  hun- 
dredth of  an  inch  for  a  pitch  of  three  inches,  a  quantity  hardly 
worth  considering  in  practice,  except  in  very  large,  heavy  gears  with 
machine-dressed  teeth. 

The  true  epicycloidal  face  for  Cab  lies  outside  the  circle  be- 
tween C  and  <?,  and  inside  between  a  and  b,  and  differs  from  the 
Willis  circular  face  in  that  the  latter  is  more  nearly  parallel  to  the 


162  PRINCIPLES   OF   MECHANISM. 

epicycloid  from  b  to  a  and  leaves  a  fullness  at  O.  In  action,  the 
Willis  teeth  will  thus  be  more  inclined  to  receive  the  heavier  bearing 
pressures  near  C  and  the  lesser  near  b,  or  have  the  working  contact 
between  the  teeth  near  the  line  of  centers  where  the  slipping  action 
is  least,  rather  than  at  a  remote  point  from  the  line  of  centers  with  a 
greater  rate  of  slip. 

By  some  it  is  thought  advisable  to  arbitrarily  allow  the  teeth  to 
be  somewhat  slack  near  ab,  to  enable  the  teeth  to  perform  with  com- 
paratively light  working  pressures  here,  and  heavier  at  or  near  C, 
thus  reducing  the  strains,  the  friction,  and  the  wear  of  the  teeth. 
To  this  end,  the  radii  from  the  Grant  tables  may  be  arbitrarily  re- 
duced in  length  to  give  a  corresponding  slackness  in  the  neighbor- 
hood of  ab. 

In  Fig.  186,  the  two  points  d  and  e,  in  the  flank,  chosen  to  lo- 
cate the  flank  circle  arc,  provide  for  a  greater  depth  of  flank  than 
ever  goes  into  action  on  a  face,  and  it  seems  probable  that  e  should 
have  been  chosen  noc  much,  if  any,  below,  the  working  depth  near 
the  middle  of  Ce,  in  order  to  the  greatest  precision  of  the  actual 
working  tooth  profile.  As  a  result,  the  circle  arc  between  C  and  d 
makes  the  tooth  fuller  than  would  the  actual  hypocycloid. 

If  an  "  odontograph  "  is  to  mean  an  instrument,  it  appears  that 
the  Grant  odontograph  is  simply  any  ordinary  measuring  scale. 
Thus  the  pocket  rule  becomes  an  odontograph  which  by  analogy 
with  the  Willis  odontograph  is  used  in  determining  two  distances 
on  the  drawing,  instead  of  one  angle  and  one  distance. 

Circle- Arc  Tooth  Outlines  may  be  determined  directly  from  the 
rolled  epicycloids  or  involutes;  as,  for  instance,  in  Fig.  178,  having 
drawn  the  addendum  circle  and  supposing  it  to  have  cut  near  G, 
find  by  trial  a  center  point  that  will  give  a  circle  arc  that  approxi- 
mates most  favorably  with  the  curve  CFG.  Then,  noting  that 
center  point  and  radius,  describe  a  circle  through  the  point  concen- 
tric with  the  pitch  line,  and  make  it  the  locus  of  the  centers  of  all 
tooth-face  circle  arcs  for  that  wheel,  and,  using  the  radius  found 
above,  strike  in  the  tooth  faces.  Likewise  proceed  for  the  flanks. 
Involute  profiles  require  but  one  center  and  radius. 

Co-ordinating  of  the  Tooth  Profiles  has  been  worked  out  very 
completely  by  Prof.  J.  F.  Klein  of  Lehigh  University,  whereby  the 
correct  tooth  profile  curve  is  plotted  by  aid  of  co-ordinates  taken 
from  a  table. 

This  method  would  seem  to  possess  special  advantages  for  large 
teeth  where  accuracy  is  required  and  where  the  ordinates  have  a 


PRACTICAL   CONSIDERATIONS.  163 

length  convenient  to  lay  off.  For  teeth  of  one  inch  pitch  or  less 
a  magnifying-glass  would  seem  to  be  a  necessity,  as  well  as  very 
fine  measuring  and  drawing  devices.  Very  elaborate  tables  accom- 
pany the  method  and  are  embraced  in  Klein's  Elements  of  Machine 
Design. 

These  Various  Methods  of  drawing  the  teeth  have  each  their 
peculiar  advantages,  the  most  advisable  for  one  case  not  being  so 
for  some  others.  For  instance,  when  very  large  teeth  are  to  be 
drawn,  with  plenty  of  time  for  accuracy,  the  designer  may  do  dif- 
ferently than  when  the  teeth  are  more  moderate  in  size  and  more 
hurried. 

All  of  the  above,  however,  admits  of  more  or  less  of  "hand  and 
eye  "  operations,  in  which  errors  of  greater  or  less  magnitude  will 
be  unavoidable. 


CHAPTER  XIV. 


MACHINE-MADE  TEETH. 

GEAR  teeth,  where  no  "  hand  and  eye  "  process  enters  the  ac- 
count, are  possible,  and  in  fact  are  in  practical  commercial  operation 
to-day,  both  where  the  processes  conform  strictly  with  theory,  and 
where  they  only  approximate  it.  Doubtless  watch  gearing  is  the 
best  example  of  the  former,  the  hand  and  eye  operations  all  being 
eliminated,  as  well  they  may  be  from  the  very  fact  of  the  minute- 
ness of  the  teeth. 

In  Fig.  187  is  illustrated  what  may  be  termed  an  Epicycloidal 
Machine  used  to  cut  a  tool  to  the  truly  epicycloidal  shape. 
Here  the  tool  D  to  be  formed  epicycloidal  is  made  fast  to  the 


FIG.  187. 

shaft  G.  One  end  of  the  shaft  is  pivoted  by  universal  joint  to  the 
post  F,  while  the  other  end  has  the  disk  A  made  fast  to  it.  This 
disk  rolls  upon  the  stationary  disk  B.  At  E  is  a  guiding  wheel 
charged  with  diamond  dust  or  emery. 

The  disk  A  represents  the  pitch  circle  of  the  wheel  for  the  teeth 
of  which  the  tool  D  is  to  be  shaped,  while  the  disk  B  represents 
the  pitch  circle  of  the  pinion. 

164 


MACHINE-MADE   TEETH. 


165 


As  A  rolls  upon  B,  the  tool  blank  D  is  made  to  rub  against 
the  revolving  grinding  wheel  E.  By  repeated  rolling  movements 
of  A  upon  B}  the  former  being  shifted  occasionally  to  bring  D  to 
touch  the  grinder  E,  the  tool  will  finally  become  ground  so  as  to 
touch  E  slightly  throughout  the  rolling  of  A  upon  B,  when  the 
tool  D  will  be  finished  to  the  true  epicycloidal  shape. 

The  pivoting  of  one  end  of  the  shaft  or  bar  G  at  F  has  the  ef- 
fect of  making  A  and  B  the  bases  of  rolling  cones  with  common 
vertex  at  F,  so  that  by  passing  a  plane  transversely  at  the  tool  D 
we  cut  the  cones  in  circles  proportional  to  the  circles  A  and  B. 
Thus,  the  nearer  the  tool  is  to  F  the  more  minute  in  effect  will  the 
rolling  circles  become,  so  that  with  D  at  the  proper  point  on  G 
our  rolling  circles  will  be  of  watch-wheel  dimensions,  while  the 
circles  A  and  B  will  be  large  enough  to  manipulate  conveniently. 

To  prove  that  we  thus  obtain  the  truly  epicycloidal  form  of 


FIG.  188 

ground  curve  on  the  jend  of  D,  we  refer  to  Fig.  188,  which  repre- 
sents a  section  taken  at  D,  through  the  tool  and  the  cones  A  and 
B.  Here,  however,  A  and  the  tool  are  supposed  stationary,  with  B 
rolling  upon  it,  and  with  the  grinding  disk  E  also  moving  around 
with  B.  In  Fig.  187,  the  grinder  E  is  placed  with  its  face 
coincident  at  the  axis  of  the  cone  B,  as  shown  in  the  section,  Fig. 
188.  Then,  supposing  E  to  move  with  B  as  the  latter  rolls  on  A, 


166  PRINCIPLES    OF    MECHANISM. 

and  noting  them  in  several  positions  as  at  JJT,  etc.,  we  find  that 
the  face  of  E  wipes  up  an  epicycloid  DJK,  the  same  as  if  a 
describing  circle,  CL,  were  used  with  diameter  equaling  the  radius 
of  Bf  and  with  a  tracing  point  at  L.  Thus  the  end  of  D  has  the 
epicycloidal  curve  DJ  formed  upon  it,  with  a  continuation  from  D 
to  H  on  a  radius  of  A.  This  line,  HDJ,  is  perceived  to  be  exactly 
that  for  a  tooth  profile  for  A,  answering  to  the  case  of  flanks 
radial  in  both  pitch  lines  B  and  A.  The  relative  as  well  as  ab- 
solute sizes  of  A  and  B  are  seen  from  the  preceding  to  be  entirely 
arbitrary,  so  that  profiles  HDJ  may  be  obtained  for  a  wheel  or  for 
a  pinion. 

The  tool  D  thus  obtained  is  only  provisional  to  the  final  "  fly 
cutter"  for  cutting  the  watch  wheels,  or  the  little  milling  cutter 
for  the  pinions;  and  its  application  is  shown  in  Fig.  189,  where  D 
is  used  as  a  planing  or  turning  tool,  as  the  case  may  be,  to  form  /, 
the  final  cutter  for  cutting  the  wheel  A. 

The  above  process  is  much  simplified  by  the  fact  of  radial 
flanks  for  the  teeth.  These,  for  watches, 
have  ample  strength  and  are  light-run- 
ning in  action. 

The  same  process  may  be  applied  for 
wheels  of  any  size,  and  when  for  ordin- 
S      /     ary  machinery,  cylindrical  wheels,  A  and 
ff     Bt  may  be  used  instead  of  the  conical 
ones  in  Fig.  187,  though  for  cones  AF 
and    BF,  of    considerable    length    and 
gradual   convergence,   the   error  due  to 
the  cones  may  be  reduced  to  an  inap- 
preciable  quantity,   with   results   which 
FIG.  189.  are  practically  perfect.     Teeth,  as  above, 

are  independent  of  all  "  hand  and  eye  " 

processes  such  as  locating  an  irregular  curve  to  draw  a  tooth 
face,  filing  a  sheet  zinc  templet  to  fit  a  tooth  profile,  etc. 

The  teeth  as  produced  above,  all  having  radial  flanks,  though 
entirely  suited  to  such  special  sets  of  gearing  as  used  in  watches, 
clocks,  and  some  heavier  machinery,  where  interchangeability  is  not 
sought,  are  not  adapted  for  outfits  of  simple  sets  of  cutters  for 
general  commercial  purposes,  either  in  the  tool  store  or  in  the 
machine  shop.  For  this  it  is  generally  admitted  that  there  are  but 
two  systems  of  gearing  and  cutters  in  use,  viz : 


MACHINE-MADE   TEETH.  167 

The  Involute  and  the  Epicycloidal  Cutters. 

These,  for  cutting  interchangeable  gearing  of  its  own  kind  are 
in  extended  use:  the  first  in  accordance  with  principles  explained 
in  Fig.  166,  and  the  second  following  the  principles  of  Fig.  153. 

In  either  of  these  systems,  cutters  are  made  in  series,  each  of 
which  will  cut  several  wheels  of  differing  tooth  numbers,  one  of 
which  gears  will  be  right,  the  others  slightly  inaccurate,  but  none 
out  of  the  theoretic  condition  to  an  extent  to  produce  appreciably 
bad-working  wheels. 

The  first  system  of  cutters  of  this  kind  for  cutting  gear  wheels 
were  put  out  by  the  Brown  &  Sharpe  Mfg.  Co.,  and  were  approx- 
imately involute  in  form  of  tooth  cut,  each  pitch  embracing  eight 
cutters  to  cut  gears  of  from  twelve  teeth  to  a  rack,  and  inter- 
changeable— one  cutter  cutting  gears  of  twelve  and  thirteen  teeth, 
the  next  fourteen  and  sixteen,  the  next  seventeen  to  twenty,  the 
next  twenty-one  to  twenty-five,  the  next  twenty-six  to  thirty-four, 
the  next  thirty-five  to  fifty-four,  the  next  fifty-five  to  one  hundred 
and  thirty-four,  the  last  one  hundred  and  thirty-four  to  a  rack. 

The  second  system  of  cutters  for  epicycloidal  teeth  of  wheels  in 
interchangeable  series  are  also  made  by  the  Brown  &  Sharpe  Mfg. 
Co.,  but  by  machinery  brought  out  by  the  Pratt  &  Whitney  Co. 
This  system  embraces  twenty- four  cutters  for  each  pitch,  cutting 
from  twelve  teeth  to  a  rack,  a  wheel  of  fifteen  teeth  having  radial 
flanks.  For  a  full  description  of  the  machinery  for  making  these 
cutters,  with  no  "hand  and  eye"  operation,  see  MacCord's  Me- 
chanical Movements,  page  178. 

There  are  two  machines,  one  of  which  is  called  the  Epicycloidal 
Milling  Engine,  which  mills  or  cuts  out  all  the  tooth  templets  re- 
quired in  the  series,  of  magnified  size;  and  the  other  is  the  Panta- 
graphic  Cutter  Milling  Engine,  which  applies  the  above  templets 
in  making  the  gear-tooth  cutters  reduced  to  any  desired  size  or 
pitch.  Other  templets  may  also  be  used  on  this  last-named 
engine. 

The  number  of  cutters  in  a  series  is  arbitrary  and  depends  upon 
the  inaccuracy  allowed  to  be  admissible.  For  large  cutters,  as  for 
large  teeth,  it  would  seem  that  the  number  of  cutters  in  a  series 
should  be  relatively  great,  that  the  admitted  errors  of  cut  teeth 
may  not  exceed  a  certain  arbitrary  value.  A  formula  for  deter- 
mining the  so-called  "equidistant  series"  of  cutters  is  given  by 
Geo.  B.  Grant  in  Teeth  of  Gear  Wheels,  page  20.  Experience  in 


168 


PRINCIPLES   OF   MECHANISM, 


the  management  of  gearing  would  lead  one  to  adopt  a  tooth 
numbering  in  the  above  series  of  cutters  such  that  the  cutter  that 
cuts  gears  from  50  to  60  teeth,  for  instance,  will  make  the  50- 
toothed  wheel  right  and  the  others  slack  on  the  point,  rather  than 
the  contrary,  that  the  teeth,  in  action,  may  have  the  severest  pressure 
of  contact  near  the  line  of  centers  rather  than  remote  from  it,  as 
explained  at  Fig.  186. 

Prof.  Edward  Sang's  Theory  of  the  Conjugating  of  the  Teeth 
of  Gear  Wheels  in  interchangeable  series  has  been  applied  in  two 
different  and  patented  gear-cutting  engines,  the  first  by  Ambrose 
Swasey  of  Cleveland,  Ohio,  Pat.  No.  327,037;  and  the  other  by  Geo. 
B.  Grant  of  Lexington,  Mass.,  Pat.  No.  405,030. 

In  the  first,  or  the  Swasey  Engine,  a  split  multiple  cutter  is 
employed  to  cut  the  gear  teeth  directly;  while  in  the  second  a 
solid  worm  or  screw  hob  is  used  in  cutting  the  gear.  In  each  of 
these  one  and  the  same  cutter  cuts  all  gears  of  a  given  pitch,  re* 
gardless  of  size  of  wheel,  and  giving  theoretically  correct  teeth  for 
all  without  the  intervention  of  hand  and  eye  operations. 

Let  Fig.  190  represent  Sang's  principle  in  the  case  of  Fig.  153, 
as  applied  to  the  rack  where  the  de- 
scribing circles  for  face  and  flank 
are  one  and  the  same,  carrying  the 
tracer  D  to  trace  the  face  CD,  and 
the  tracer  E  to  trace  the  flank  CE  of 
the  rack  tooth  as  the  describing  circle 
rolls  along  the  straight  pitch  line. 
These  curves  for  face  and  flank  are 
both  cycloids  and  identical  in  form. 

Now,  as  any  wheel  of  the  interchangeable  series  of  Fig.  153,  of 

the  same  pitch  and  describing 
circle,  will  work  correctly  with 
this  rack,  it  is  plain  that  if  a 
multiple  cutter  were  made  with 
cutting  teeth  of  the  same  longi- 
tudinal section  as  Fig.  190,  that 
cutter  would  cut  any  gear  of  tho 
series  referred  to,  if,  while  cut- 
ting, the  cutter  could  be  moved 
relatively  to  the  gear,  as  A  is 
FIG.  191.  relative  to  B  in  Figs.  123  and 

124;  or  as  shown  in  Fig.  191,  where  D  is  the  multiple  cutter,  and 


FIG.  190. 


MACHINE-MADE   TEETH.  169 

/ 

B  the  wheel  being  cut  while  it  moves  along  as  if  its  pitch  line  were 
rolling  on  the  pitch  line  EF  from  E  to  ff  the  cutter  at  the  same 
time  revolving  and  cutting. 

This  would  seem  to  require  a  cutter  as  long  as  the  circumference 
of  the  gear.  To  avoid  this,  Mr.  Swasey  splits  the  cutter,  D,  into 
halves,  so  that  the  idle  half  can  move  back  a  pitch  while  the  other 
half  is  cutting  and  moving  with  the  periphery  of  the  wheel,  the 
latter  being  kept  in  steady  revolving  motion  with  its  axis  stationary. 
In  this  way,  the  cutter,  D,  may  be  so  short  as  to  reach  from  the  in- 
tersection of  its  addendum  line  with  that  of  the  describing  circle 
on  one  side  over  to  like  intersection  on  the  other  side,  or  from  G 
to  H,  Fig.  190. 

In  cutting  a  gear,  B,  the  latter  is  kept  steadily  revolving,  and 
also  the  cutter  D,  in  exact  relation  by  gearing,  the  cutter  making 
as  many  revolutions  to  one  of  the  gear  blank  to  be  cut  as  there 
are  to  be  teeth  in  that  wheel.  Then  with  B  and  D  in  motion  in 
the  cutting  engine,  a  slow  feed  is  given  to  the  cutter,  D,  toward  the 
wheel  blank,  B,  cutting  all  the  teeth  together  by  one  continuous 
action.  When  the  gear  is  partly  cut,  all  the  teeth  are  cut  to  the 
same  extent,  and  all  are  finally  finished  at  the  same  time. 

To  form  the  cutter  D,  Fig.  191,  a  tool  is  required  which,  on  its 
cutting  end  has  the  exact  shape  of  a  space  in  the  rack,  or  of  ICJKNL, 
Fig.  190.  This  tool  may  best  be  formed  by  some  such  device  as 
the  Pratt  &  Whitney  Pantagraphic  Cutter  Milling  Engine  men- 
tioned above,  if  it  is  to  have  the  cycloidal  form  without  hand  and 
eye  processes. 

But  in  the  Bang's  theory  we  are  not  confined  to  the  cycloidal 
form  of  curves  ICJ  and  KNL\  for  a  brief  consideration  will  show 
that  any  arbitrary  curve  1C  may  be  adopted  and  copied  at  CJ  by 
retaining  the  point  C  in  common  and  swinging  /  around  to  J  in  the 
plane  of  the  paper,  and  then  the  whole  curve  ICJ  turned  over  and 
copied  at  KNL.  Thus,  for  the  pantagraphic  milling  engine,  a  circle 
may  be  cut  in  a  lathe  for  the  parts  1C  and  CJ  of  the  templet. 

Again,  and  for  the  simplest  case  under  Sang's  theory,  1C  may 
be  a  straight  line,  which  should  be  located  at  a  certain  angle  of  IJ 
with  NC,  usually  about  14 \  degrees.  Also  the  same  for  KL. 

This  last-named  case  gives  us  the  well-known  series  of  inter- 
changeable wheels  with  involute  teeth. 

This  multiple  cutter,  thus  made  in  two  parts,  is,  in  effect,  turned 
in  a  lathe,  by  cutting  one  space  groove  after  another  with  the  tool 


170 


PRINCIPLES   OF    MECHANISM. 


formed  as  above  explained,  the  top  face  of  the  tool  being  held  in  the 
meridian  plane  of  D  while  cutting  it. 

In  the  Grant  Cutting  Engine  the  splitting  of  the  cutter  Z),  thus 
complicating  and  weakening  it,  is  avoided  by  the  use  of  a  solid 
worm  cutter  or  hob,  the  latter  making  one  revolution  while  the 
gear  turns  one  pitch.  In  cutting  a  gear,  the  motions  of  the  cutter 
and  gear  bank  are  continuous  and  in  definite  relation  till  the  gear 
is  finished,  similarly  as  in  the  Swasey  machine,  and  this  machine 
would  seem  to  have  the  preference,  unless  it  is  found  that  the  hob 
cutter  is  so  complex  in  form  of  worm  threads  as  to  be  unduly  diffi- 
cult to  make. 

Investigating  this,  we  find  that  hobs  for  cutting  epicycloidal 
teeth  must  have  the  axes  of  the  hob  inclined  to  the  plane  of  the 
gear  being  cut  by  an  angle  equal  that  between  a  plane  normal 
to  the  axis  of  the  hob  and  the  tangent  to  the  worm  thread  of  hob, 
at  the  pitch  line.  That  is,  the  element  of  thread  of  hob,  at  the 
pitch  line,  must  be  perpendicular  to  the  plane  of  the  wheel.  This 
inclination  can  easily  be  found  from  a  triangle  where  one  side  is  the 
circumference  of  the  hob  at  the  pitch  line,  another  side,  perpendic- 
ular to  the  first,  the  pitch;  in  which  triangle  the  smaller  angle  is  to 
be  taken  for  the  inclination. 


FIG.  192. 

Even  with  this  inclination  of  the  thread  of  the  hob  (speaking  of 
it  as  before  the  teeth  are  cut  in  it  and  when  it  is  simply  a  screw),  it 
will  not  be  tangent  to  the  cut  teeth  of  a  gear,  or  of  a  rack,  Fig.  1 90, 
cut  by  it,  except  in  an  S-shaped  curve,  acb,  Fig.  192,  which  is  more 
pronounced  as  the  radius  r  is  shorter  and  not  vanishing  until  r  Is 
infinite.* 


*  The  equation  of  this  curve  acb  is  r  sin  9  —  r 


MACHINE-MADE   TEETH.  171 

To  give  an  idea  of  the  intensity  of  the  S-curve  for  epicycloidal 
teeth,  take  rl  =  2",p  =  0.5".  Then  db,  Fig.  192,  is  0.158";  and 
for  the  ten  equal  divisions  from  d  to  «,  the  remaining  ordinates  are 
0.149",  0.126",  0.104",  0.075,  .0",  -  0.078",  -  0.107",  -  0.129", 
—  0.147",  and  —  0.162",  the  extreme  ordinates  being  the  greater. 

The  tool  which  cuts  the  worm,  as  in  a  screw-cutting  lathe, 
should  have  this  shape  on  its  top  side  at  the  cutting  end,  as  shown 
dotted  at  G  and  H,  in  case  the  tool  is  to  be  used  as  in  cutting  other 
threads.  But  a  preferable  way  is  to  make  its  top  flat  and  straight 
and  of  the  true  epicycloidal  shape  in  plan  as  at  //,  and,  after  cutting 
in  the  ordinary  way  to  the  right  depth  for  the  hob  thread,  then, 
without  varying  the  depth  of  cut,  make  several  passes  for  cuts  with 
the  tool  at  varied  heights,  raising  and  lowering  on  a  line  which  is 
tangent  to  the  hob  thread  at  the  pitch  line,  and  for  a  range  cover- 
ing the  ordinates  db  and  ea.  A  safe  way  would  be  to  work  the  tool 
at  different  heights  thus  as  long  as  it  will  take  a  cut.  This  sup- 
poses that  the  tool  for  cutting  the  hob  thread  has  the  correct  shape 
as  at  ICJKNL,  Fig.  190,  and  formed  as  already  explained. 

But  in  cases  where  the  tooth  profiles  are  not  normal  to  the 
line  GN9  Fig.  190,  the  cutter  hob  will  not  need  inclining  so  much, 
and,  in  some  cases,  not  at  all,  as  for  the  involute  series. 

The  hob  for  involute  toothed  wheels  in  interchangeable  series  for 
use  on  the  Grant  Cutting  Engine  is  to  be  made  in  a  similar  way 
if  its  axis  is  to  be  inclined  as  before,  so  that  at  the  finishing  of  cut 
the  tangent  to  the  pitch-line  element  of  the  hob  thread  is  normal  to 
the  plane  of  the  wheel  being  cut.  But  for  this  case  of  involute 
gearing  it  is  not  necessary  to  incline  the  axis,  as  it  may  be  kept 
parallel  to  the  face  of  the  wheel  being  cut,  if  the  hob  is  formed 
with  due  regard  to  this,  viz. :  that,  in  cutting  the  hob  threads,  as  in 
a  threading  lathe,  the  tool  be  held  at  varying  heights  as  before, 
while  cutting,  except  that  now  it  is  to  be  raised  and  lowered  on  a 
vertical  line  instead  of  a  tangent  to  the  pitch-line  element,  and  to 


which,  for  epicycloidal  series,  A  -  -  and  B  =  —A/ r  ~~  Tl      _  in  wnich 

f  p  *    7.5p  —  r-f-ri* 

last,  p  is  the  diametral  pitch. 

In  involute  series  with  axis  of  hob  inclined  as  above  stated,  A  =  —    and 

r  ' 

B  =  ~p  '  tan  U^  =  °52  p*     In  constructin£  tliese  curves,  r  is  to  be  laid 

off  from  D  on  DC  ;  and  r  sin  0  is  to  be  laid  off  from  DC  on  circle  arcs  struck 
from  D. 


172  PRINCIPLES    OF   MECHANISM. 

•conthme  the  raising  or  lowering  and  repeated  cuts  as  long  as  it  will 
take  cuts.  In  this  case,  however,  the  tool  may  be  placed  at  the 
height,  first  above  and  afterwards  below  the  axes  of  the  hob,  in 
cutting  the  thread  equal  the  diametral  pitch  divided  by  twice  the 
tangent  of  the  angle  of  inclination  of  the  edges  of  the  tool,  or  of 
the  sides  of  a  tooth  in  the  rack,  which  inclination  is  usually  about 
14£  degrees.  Hence  the  height  above  or  below  is  about  twice 
the  diametral  pitch. 

In  all  cases  above,  the  thread  tool  for  cutting  the  hob  threads  is 
to  be  correct  in  shape  and  dimensions  as  answering  to  the  outline 
1CJKNL,  Fig.  190,  except  that  it  should  be  somewhat  in  excess  of 
length  so  as  to  cut  the  hob  deep  enough  to  have  clearance  at  the 
top  of  the  teeth,  and  with  due  regard  to  side  clearance  between  re- 
sulting teeth  of  wheel. 

It  appears  that  the  hob  threading  tool  must  have  the  correct 
.shape  in  any  event  according  to  the  system,  and  of  the  right  thick- 
ness for  a  tooth;  but  that  the  threads  of  the  hob  are  not  of  correct 
tooth  profile  shape  in  any  case.  For  involute  teeth  these  threads 
will  not  be  straight  on  the  meridian  intersections.  Hence  teeth 
of  spur  wheels  cut  in  this  way  in  the  Grant  Cutting  Engine  with 
the  ordinary  worm-wheel  hobs  will  not  be  correct  in  profile  of 
tooth. 

In  these  gear-cutting  engines  of  the  Swasey  or  Grant  order 
the  spacing  of  the  teeth  may  be  expected  to  be  unusually  even  and 
exact,  since  in  the  cutting  the  hob  acts  upon  several  teeth  at  the 
same  time,  and  the  wheel  being  cut  may  be  driven  around  by  worm 
and  wheel.  The  latter  may  be  made  in  two  half  wheels  joined  on 
a  plane  transverse  to  the  axes  and  through  the  middle  of  the  teeth. 
In  cutting  this  duplex  worm  wheel,  when  partly  cut  it  may  have 
one  half  loosened  and  shifted  on  the  other  half  the  space  of  a  few 
teeth,  made  fast  and  some  further  cutting  done,  then  shifted  again 
and  again,  till  the  teeth  in  the  two  halves  will  all  fit  exactly  for 
any  way  of  combining  the  halves.  Then  when  in  use  and  wear  the 
part  may  be  occasionally  shifted.  In  this  way,  with  these  cutting 
machines,  it  seems  certain  that  the  spacing  as  well  as  the  tooth 
outline  of  cut  gears  may  be  made  marvels  of  exactness. 

Tooth  Planing  or  Dressing  Machines. 

These  machines,  like  the  Gleason's,  are  in  use  in  some  machine- 
shops,  where  the  teeth  of  large  heavy  cast  gears  are  tool  dressed,  as 


MACHINE-MADE  TEE^H.  /,f  J    173 

^^CAUFORH^^ 

in  a  shaping  machine,  the  tool  being  guided  by  a  templet  which 
may  have  been  formed  by  hand,  thus  admitting  hand  work  in  part. 

George  H.  Corliss  appears  to  have  been  the  first  to  do  this,  his 
first  work  dating  back  to  the  40's  or  50's:  the  heavy  fly-wheel 
gear  on  the  Centennial  Corliss  Engine,  central  in  Machinery  Hall, 
being  one  of  the  more  notable  examples  of  tool-dressed  spur  wheels. 
Large  bevel  wheels,  connecting  the  main  shafting  with  the  above 
engine  were  also  tool  dressed  on  a  bevel  "  gear  planing  "  machine 
exhibited  by  Mr.  Corliss. 

Hugo  Bilgram  exhibited  remarkably  smooth-running  bevel- 
gears  at  the  World's  Fair  of  1893,  the  teeth  of  which  were  dressed 
out  in  a  similar  way,  as  an  example  of  commercial  work  by  him. 

In  these  tooth-planing  machines  for  bevel  gears  the  point  of 
the  cutting  tool  is  made  to  move  on  a  slide  in  a  line  joining  the 
vertex  of  the  cone  of  the  gear  and  the  point  of  contact  of  the  guide 
finger  with  the  guiding-tooth  templet,  this  tool  and  finger  being 
mounted,  in  effect,  on  a  universally  swinging  arm,  pivoted  at  the 
cone  vertex.  Thus  the  elements  of  the  finished  tooth  are  made  to 
converge  to  the  apex  of  the  conic  gear.  Large  or  small  gears  can 
be  tooth  dressed  on  the  same  machine.  One  considerable  advan- 
tage here  over  the  spur-gear  tooth-dressing  machine  is  that  the 
tooth  templet  may  be  made  of  magnified  size,  so  that  bevel  wheels, 
compared  with  spur  wheels  thus  tooth-dressed,  may  be  regarded  as 
more  nearly  perfect,  while  the  contrary  is  true  of  wheels  cut  in 
the  ordinary  way  with  revolving  cutters. 

Stepped  and  Spiral  Spur  Gearings. 

In  some  of  the  finer  light-running  machinery  at  high  speed  the 
plain  spur  gearing  is  quite  likely  to  make  a  humming  sound  as  the 
teeth  engage  in  succession,  each  contact  giving  a  slight  click. 

This  is  avoided  in  a  measure  by  stepped  gears,  the  wheel  being 
made  up  of  several  thin  ones  made  fast  together  and  arranged  in 
steps  so  as  to  divide  the  pitch  into  as  many  parts  as  there  are  thin 
parts  in  the  wheel. 

Another  way  is  to  use  the  spiral  gearing  of  Hooke,  in  which 
each  tooth  is  on  a  spiral  slant,  to  such  extent  that  one  pair  of  teeth 
engage  before  the  preceding  pair  quits  engagement. 

This  causes  endlong  pressure  on  the  axes,  objectionable  in 
heavy  working  wheels,  but  which  is  often  obviated  by  making  each 
tooth  of  equal  portions  of  right  and  left  handed  spirals. 


CHAPTER  XV. 
SECOND:    AXES   MEETING. 

BEVEL   GEARING. 

THE  teeth  of  these  wheels  may  be  made  of  any  of  the  forms  ex- 
plained under  the  case  of  axes  parallel.  The  pitch  surfaces  being 
cones,  with  vertices  at  a  common  point,  the  describing  curves,  by 
analogy  with  cylindric  gears,  are  also  cones,  with  vertices  in  com- 
mon with  the  pitch  cones. 


Theoretically  Correct  Solution. 

In  practice  the  bases  of  the  cones  may  be  made  spherical,  with 
centers  at  the  cone  vertices,  and  the  describing  cones  may  be 
realized  in  concave  templets  fitting  these  spherical  surfaces,  as 
explained  in  Fig.  138.  In  this  way,  the  correct  tooth  curves  may 
be  laid  out  on  the  gear  blanks,  the  number  and  kinds  of  templets 
required  being  explained  in  Figs.  Ill  and  112,  though  here  all 
made  concave  and  fitting  the  spherical  base  of  the  cone. 

In  these  wheels,  no  circular  rolling  cone  can  describe  precisely 
radial  flanks,  and  to  approximate  them  the  describing  circular 
cone  rnnst  be  made  a  trifle  larger  than  to  span  the  distance  on  the 
sphere  from  the  pitch  line  to  the  pole  or  axis  to  the  wheel. 

Approximate  Solution. 

But  a  much  simpler  way,  and  that  usually  followed  in  practice, 
is  Tredgold's  approximate  construction,  which,  though  slightly 
inaccurate  in  theory,  is  appreciably  exact  in  practice.  In  Fig.  193, 

174 


AXES    MEETING. 


175 


FIG.  193. 


take  A  and  B  for  the  wheels  with 
pitch  surfaces  in  contact  at  CG,  they 
being  of  conical  form,  with  the  cone 
vertices  at  a  common  point  in  0,  the 
axes  being  OAD  and  QBE.  The 
length  CG  is  arbitrary,  and  also  the 
angle  of  intersection  of  the  axes  at  0. 
In  practice  the  latter  is  usually  a 
right  angle. 

Tredgold  draws  a  line,  DCE,  per- 
pendicular to  the  line  or  element  of 
contact  CO  of  the  pitch  cones,  when, 
if  CD  be  revolved  about  the  axis  A  0, 

a  conic  surface  ACDJ  will  be  generated,  which  is  normal  to  the 
pitch  cone,  ACOJ.  Likewise  will  the  line  CE  generate  the  cone 
BCEK,  normal  to  the  pitch  cone  BCOK.  Developing  these  cones 
from  the  line  ED,  we  obtain  the  circles  of  development  DCF  and 
ECH,  with  centers  at  D  and  E. 

Upon  these  circles  the  teeth  are  laid  oufc  as  if  they  were  the 
pitch  lines  of  a  pair  of  gears,  any  system  of  teeth  being  selected 
as  pref  erred.  For  epicycloidal  teeth  with  concave  flanks,  the 
describing  circle  carrying  the  tracer  J  has  a  diameter  which  is  less 
than  CE,  and  in  the  usual  way  is  rolled  inside  of  CH  and  outside 
of  CF,  to  generate  a  pair  of  mating  tooth  curves,  a  face  and  flank. 
Another  describing  circle  is  rolled  on  the  other  sides  of  the  circles 
CF  and  Gil,  thus  completing  the  tooth  profiles  as  dotted. 

The  addendum  circles,  root  circles,  clearing  curves,  etc.,  may 
be  drawn  in,  as  in  spur  gearing,  when  the  conic  surfaces  CDFdiid 
CEH  may  be  re-developed  or  returned  to  the  cones  CDAJ  and 
CEBK,  taking  the  tooth  drawings  with  them,  giving  us  the  laid-out 

teeth  on   the  conic  blanks,  as 
in  Fig.  194. 

In  preparing  the  blanks, 
they  should  be  left  larger  to 
include  the  addenda  and  other 
portions  of  the  finished  wheel 
as  shown  in  Fig.  194,  unless  it 
is  preferred  to  dress  the  flank 
to  the  root  surface  and  add 
the  teeth  thereto  as  often  done. 


The  tooth-surface  elements  should  all  vanish  at  0,  and  a  convenient 


176  PRINCIPLES    OF   MECHANISM. 

as  well  as  sure  way  to  give  the  teeth  the  right  directions  in  the 
finished  wooden  gear  patterns,  for  instance,  is  to  fix  a  fine  line  at 
the  point  0,  which  may  be  drawn  to  the  tooth  outline  at  C  to 
determine  when  the  tooth  is  dressed  to  the  proper  lines,  element 
by  element. 

Bevel  wheels  are  not  practicable  in  interchangeable  series, 
because,  if  one  pair  have  axes  at  right  angles,  the  substitution  for 
one  of  these  of  another  correct  working  wheel  of  larger  or  smaller 
radius  changes  the  angle  between  the  axes  to  something  other  than 
the  usual  90  degrees.  Therefore  the  teeth,  and  both  wheels  entire, 
must  be  made  in  pairs  and  of  shapes  to  suit,  regardless  of  inter- 
changeability. 

Spiral  Bevel-wheel  Teeth. 

These  are  possible,  but  not  common,  because  difficult  to  make, 
except  by  special  machinery,  which  probably  does  not  exist.  Stepped 
teeth  would  be  more  readily  made. 

Bevel -wheel  teeth,  carefully  planed  to  shape  by  templet  in  a  gear- 
planing  machine,  will  work  fairly  well.  See  Tooth  Planing,  under 
Axes  Parallel. 


CHAPTER  XVI. 
THIRD:   AXES   CROSSING  WITHOUT   MEETING. 

SKEW-BEVEL  GEAR  WHEELS.    APPROXIMATE  CONSTRUCTION. 

As  stated  in  connection  with  non-circular  skew  bevels,  a  method 
of  laying  out  theoretically  correct  and  practicable  teeth  is  not 
known,  except  for  gears  taken  at  or  near  the  common  perpendicular 
between  the  axes,  or  near  the  gorge  circles. 

But  teeth  which  approximate  the  epicycloidal  form  may  be 
generated  by  employing  generating  hyperboloids  to  roll  upon  the 
pitch  hyperboloids  inside  and  outside,  in  a  manner  analogous  to  the 
use  of  describing  cones  in  common  bevel  wheels,  and  as  shown  in 
Fig.  138.  These  approximate  tooth  curves  and  surfaces  become 
more  and  more  inaccurate  as  the  angle  between  the  axes  increases 
toward  90  degrees,  at  which  limit  considerable  interference  occurs, 
requiring  the  teeth  to  be  "  doctored  "  by  arbitrarily  dressing  off 
certain  tooth  faces  or  mating  flanks  to  an  appreciable  extent,  to  get 
the  wheels  to  work  with  acceptable  smoothness. 

The  patterns  for  cast  gears  may  thus  be  executed  to  better  ad- 
vantage, probably,  than  in  any  other  way,  even  not  excepting  Prof. 
MacCord's  construction,  based  partly  on  Olivier's  theory  of  invo- 
lutes for  one  tooth  surface,  assuming  another,  and  determining  the 
rest  by  difficult  conjugating. 

To  construct  these  approximate  skew  bevels,  let  PRLNSQ,  in 
Fig.  195,  represent  the  blank  of  a  skew-bevel  wheel,  for  which  the 
hyperboloid  of  revolution  1TUVKM  is  the  pitch  surface,  deter- 
mined as  in  Fig.  10,  extended  from  IM  to  and  past  the  gorge  circle* 
TK,  reaching  UVat  a  distance  OZ '  =  OA  beyond  the  gorge  circle. 
The  addendum  and  dedendum  surfaces  PQ  and  RW  are  drawn 
in  throughout,  being  hyperboloids  of  revolution,  because  a  top  cen- 
ter line  of  an  extended  tooth,  as  well  as  a  root  line,  being  straight, 
would,  in  revolving,  sweep  up  hyperboloids  of  revolution,  for  the- 
same  reason  as  would  the  element  CO  of  the  pitch  surface.  At. 
JK  these  three  lines  are  shown  as  parallel  to  the  axis  in  projection^ 
which  determines  the  height,  ab,  of  a  tooth  at  the  gorge  circle,, 

177 


178 


PRINCIPLES   OF   MECHANISM. 


Then  the  contour  lines  of  the  addendum  surface  can  be  drawn  in 
as  hyperbolas,  NSbd  and  LYae.  The  lines  NL  and  SY  should  be 
extended  normal  to  MK,  to  the  bottom  of  the  web  upon  which  the 
teeth  are  mounted.  With  this  much  drawn,  the  skew-bevel  blank 
can  be  turned  up,  arms  and  hub  being  assigned  at  will. 


N 


FIG.  195. 

Tredgold's  method  is  adopted,  as  in  Fig.  193,  of  developing  in 
IF  the  normal  cone  (normal  to  the  pitch  surface)  upon  which  to 
lay  out  the  teeth  at  F.  The  epicycloidal  or  involute  form  of  tooth 
may  be  chosen  here,  as  well  as  in  ordinary  bevel  gearing,  but  this 
example  will  be  carried  out  in  the  approximate  epicycloidal  form, 
with  a  view  to  testing  the  theory  of  the  generating  hyperboloids. 

The  rolling  circle  applied  here  gives  a  close  approximation  to 
the  normal  section  of  a  tooth  at  7,  as  would  be  developed  by  the 
describing  hyperboloid  above  explained;  so  the  full  drawing  of  the 
skew  tooth  will  be  first  given,  and  its  errors  afterwards  sought  out. 


AXES  CROSSING   WITHOUT   MEETING.  179 

Instead  of  the  normal  section  of  tooth,  we  require  an  oblique 
section,  as  shown  at  A,  formed  by  the  intersection  with  the  tooth 
of  the  surface  of  revolution  NLRP.  To  obtain  the  excess  of 
width  of  this  oblique  over  the  normal  section,  revolve  the  line  CO 
(same  as  CO,  Fig.  10)  about  the  axis,  till  C  falls  at  A  and  0  at  E. 
Then  revolve  the  center  line  EA  of  the  tooth  about  E,  till  A  falls 
at  6r,  where  EG  is  parallel  to  the  plane  of  the  paper,  when  EG 
appears  in  its  true  length.  A  convenient  way  to  find  G  is  to  make 
EG  equal  in  length  to  CO,  G  being  on  the  line  AD.  Then  the 
angle  OGE  is  the  angle  of  obliquity  of  the  normal  section  of  tooth 
with  the  oblique  section  at  A.  Therefore  the  width  of  the  oblique 
compared  with  the  normal  section  of  tooth  at  Fis  in  the  ratio  of 
GE  to  GO.  This  may  be  laid  off  at  several  points  in  the  height 
and  the  oblique  section  determined.  If  preferred  to  do  it  now,  the 
faces  and  flanks  may  be  arbitrarily  doctored  from  the  pitch  line, 
each  way,  to  prevent  interference,  as  explained  above  in  connection 
with  Fig.  196. 

With  the  oblique  section  determined,  as  in  the  outside  lines 
at  F,  we  may  cut  a  templet  in  thin  material  and  use  it  on  ICM,  as 
explained  in  Fig.  179,  to  delineate  all  the  teeth  on  the  pitch  line 
IM. 

Then,  in  cutting  the  teeth  to  shape,  the  oblique  direction  AE 
is  probably  best  arrived  at  by  carefully  cutting  a  thin  piece  UV  to 
all  the  tooth  outlines,  beveled  back  in  excess  from  the  tooth  profiles, 
and  mounting  it  on  the  axis  AOZ  so  that  OZ  equals  OA,  and  in 
such  position  that  the  line  CO  from  the  center  of  a  tooth  at  C 
will  strike  the  center  of  an  outline  at  H.  Then  a  thread  may  be 
drawn  at  any  time,  in  dressing  a  tooth,  from  any  point  on  H  to  the 
like  point  on  C,  and  each  element  of  C  thus  dressed  the  whole 
length  of  the  tooth  to  fit  the  line.  This  process,  repeated  for  all 
the  teeth,  completes  them  unless  when  the  mating  wheel  is  likewise 
thus  far  completed  and  the  pair  be  placed  in  running  relation  it 
be  found  that  interference  exists.  If  so,  the  teeth  are  to  be  dressed 
off  at  will,  till  interference  is  relieved,  which  completes  the  wheel. 

These  wheels  may  thus  be  made  of  any  desired  length  from  C 
toward  H.  To  give  an  idea  of  the  amount  of  interference  of  the 
epicycloidal  teeth  as  above,  Fig.  196  is  introduced,  which  was  drawn 
with  great  care  and  labor  from  a  model  like  that  described  by  Geo. 
B.  Grant  in  the  American  Machinist  for  Sept.  5,  1889,  Fig.  3.  In 
the  present  example  the  axes  are  at  right  angles,  at  a  distance  of  3.7 
inches  apart  at  the  common  perpendicular,  the  wheels  being  11  and 


180 


PRINCIPLES   OF    MECHANISM. 


6.8  inches  in  diameter  respectively,  and  with  the  velocity-ratio  of 
3  to  2.  These  tooth-curves  were  traced  with  the  true  rolling  hyper  - 
boloids. 

The  figure  is  a  transverse  section,  taken  on  a  perpendicular  to- 
the  element  of  contact,  CO,  Fig.  195,  and  at  a  distance  of  5.8  inches 


FIG.  196. 

from  0.  It  cuts  through  the  two  pitch  hyperboloids  in  curves  of 
intersection  AA  and  BB,  and  the  describing  hyperboloids  in  the 
curves  DCE  and  FCG,  which  curves  of  course  are  all  ellipses. 
Also  it  cuts  all  the  mating  tooth -curves  at  three  quarters  of  an  inch 
apart  on  the  pitch  line. 

The  describing  hyberboloids  are  both  of  one  size,  and  half  as  large 
as  the  smaller  wheel,  so  that  by  theory  the  flanks  in  the  smaller 
wheel  should  be  radial.  The  figure  shows  them  to  be  very  nearly  so. 

This  is  a  somewhat  extreme  case  of  proportions,  and  we  might 
expect  high  per  cents  of  interference.  The  figure  shows  almost  no 
interference  on  the  left-hand  side  of  the  pitch  lines  up  to  a  pitch 
of  over  two  inches;  while  on  the  right-hand  side,  for  a  pitch  of  f 
inch,  14  inch,  and  2|  inch,  we  find  the  faces  and  flanks  cutting  into 
each  othei  to  a  normal  depth  of  5,  6,  and  10  hundred ths  of  an  inch 


AXES   CROSSING   WITHOUT   MEETING.  181 

respectively,  or  about  four  per  cent,  of  the  pitch.  For  a  pitch  of 
eight  inches  the  interference  depth  reaches  15  hundredths  of  an 
inch,  or  about  two  per  cent,  of  the  pitch. 

Also,  the  figure  shows  that  below  C  the  faces  of  the  smaller 
wheel  must  be  trimmed  off  toward  the  point  and  on  the  lower  side 
for  pitches  over  about  1.5  inches,  while  the  flanks  of  teeth  of  the 
larger  wheel  above  C  seem  to  need  dressing  out  on  the  lower  sides. 
The  figure  may  thus  aid  in  determining  how  to  doctor  the  teeth  in 
a  practical  case,  for  best  results. 

One  noticeable  feature  of  these  tooth  curves  is  that,  if  they 
interfere,  they  intersect  at  the  surfaces  of  the  describing  hyperbo- 
loids,  in  the  central  positions  shown,  viz. :  on  the  lines  ECD  and 
FCG, — a  fact  also  pointed  out  by  Grant  in  1889  in  the  few  curves 
shown. 

Combined  punching  and  shearing  tools  have  had  skew  gears 
some  20"  diameter. 

EXACT  CONSTRUCTION. 

The  exact  construction  of  these  teeth  has  been  best  treated  by 
Grant  in  the  American  Machinist-  and  also  in  his  valuable  work  on 
Teeth  of  Gears,  where  the  theory  of  Olivier  and  treatment  by 
Herrmann  are  discussed. 

Olivier  appears  to  have  originated  involute  tooth  surfaces,  which 
lie  calls  spiraloids,  that  may  be  generated  by  the  revolution  of  a 
straight  line  about  a  cylinder,  the  line  being  maintained  at  a  con- 
stant angle  with  the  cylinder  and  prevented  from  lateral  slip.  The 
same  result  is  accomplished  by  rolling  a  plane  around  a  cylinder 
without  slipping,  on  which  plane  a  line  is  drawn  obliquely.  This 
line  will  sweep  up,  in  the  space  about  the  cylinder,  the  spiraloid  of 
Olivier,  which  spiraloid  surfaces  are  proposed  for  skew-bevel  teeth. 

Any  point  in  the  above  line  will  evidently  describe  an  involute 
about   the  cylinder,  and   all   points   of  the  line 
together,  will  describe  an  involute  spiral  or  spir- 
aloid. 

In  this  description  it  is  immaterial  to  the 
result  if  the  plane  slips  on  the  cylinder  in  the 
direction  of  the  line,  that  is,  if  the  line  be  only 
allowed  to  slide  on  itself,  not  laterally,  and  the 
angle  between  the  line  and  cvlinder  be  preserved 
constant. 

Fig.  197  shows  a  section  at  right  angles  to  the  spiraloid  as 


182 


PRINCIPLES    OF    MECHANISM. 


swept  up  by  one  line  on  the  plane,  CED  being  cut  by  the  line  at 
one  side  of  that  point  of  the  line  which  comes  to  touch  the  cylinder 
A  at  C,  and  CFD  cut  by  the  line  at  the  other  side  of  the  same  point. 

THE  OLIVIER  SPIRALOID. 

In  a  general  view  this  spiraloid  with  section,  Fig.  197,  appears 
like  a  twisted  bar  in  which  the  cylinder  A  C  is  straight,  the  de- 
pression C  being  like  a  spiral  or  helical 
'crease  along  the  length,  and  the  edge  D  like 
a  spiral  ridge.  According  to  what  is  stated 
above,  the  curves  of  cross-section  CED  and 
CFD,  Fig.  197,  are  the  ordinary  involutes  to 
the  circle  AC. 

In  Fig.  198  two  views  are  presented  of  a 
multiple  spiraloid  of  six  ridges  and  creases,, 
the  intersection  of  each  ridge  by  a  normal 
plane  giving  two  equal  limited  involutes,  as 
CD  and  ED.  Extended  involutes  give  Fig. 
203. 

A  straight  line,  FGH,  will  touch  one 
ridge  from  F  to  G,  while  from  G  to  H  it  lies 
in  contact  with  the  under  side  of  the  next 
ridge  above  FG.  Therefore  these  ridges 
are  enveloped  by  surfaces  which  have  in- 
volute elements  in  normal  section  and  right- 
lined  elements  in  lines  tangent  to  the  cylinder 
FIG  198  ^  CEJ,  as  in  the  example  of  the  line  FGH. 

Interchangeability  of  the  Olivier  Spiraloids. 

Now  if  we  remove  the  alternate  ridges  or  threads,  and  cut 
clearance  grooves  into  the  cylinder  to  sufficient  depth,  as  shown 
by  the  dotted  lines  UK,  this  multiple  spiraloid  will  work,  accord- 
ing to  Olivier,  as  a  skew-bevel  gear  with  another  like  it,  or  with 
any  other  of  the  same  normal  pitch,  regardless  of  the  axial  or  the 
circumferential  pitch  of  the  helical  threads  or  teeth.  This  is  true 
even  at  the  limit  where  the  helical  ridges  become  parallel  to  the 
axis,  provided  that  the  normal  pitch  is  still  the  same  and  that  the 
normal  planes  cut  the  ridges  in  involute  lines  of  section,  as  at  CDr 
Fig.  198. 

This  is  equivalent  to  saying  that  a  numerous  set  of  these  spira- 


AXES   CROSSING    WITHOUT   MEETING.  183 

loids  of  the  same  normal  pitch  will  work  together  interchangeably. 
Herrmann  pronounces  against  this. 

Interference  of  Olivier  Teeth. 

But  Olivier's  claims  can  be  shown  to  be  true  for  these  gears  as 
cut  off  at  the  intersection  of  the  axes,  as  shown  in  Fig.  199,  when 
so  cut  that  no  contacts  of  teeth  occur  outside  of  the  angle  A  OB, 
because  interference  of  teeth  takes  place  outside  these  limits,  of  the 
same  kind  as  found  to  occur  between  involute  teeth  of  cylindric 


FIG.  199.  FIG.  200. 

gears  with  involutes  drawn  from  the  pitch  circles  instead  of  base 
circles  within.     An  example  is  shown  in  Fig.  200  at  D. 

By  undercutting  the  teeth  in  the  vicinity  of  0,  Fig.  199,  and 
to  some  extent  up  the  face  CD,  Fig.  198,  with  plenty  of  bottom 
clearance,  the  teeth  will  work  when  the  gears  extend  both  ways 
past  0,  Fig.  199,  if  the  addendum  is  not  excessive,  as  has  been, 
shown  by  Oscar  Beal  in  good  working  gears  of  the  kind  in  metal. 

Nature  of  Contact  of  Olivier's  Teeth. 

Herrmann  claims  that  these  gears,  if  working  at  all,  will  have 
tooth  contacts  only  at  points  instead  of  lines,  but  it  can  be  shown 
that  the  contacts  will  be  in  lines  like  FG,  Fig.  198. 

Olivier  holds  that  the  gears  as  cut  off  at  0,Fig.  1 99,  will  drive  only 
in  one  direction,  and  that  to  drive  both  ways  they  must  be  extended 
beyond  0.  This  is  proved  by  the  Beal  gears  to  be  true,  as  well  as 
the  necessity  for  clearance  for  thick  addenda,  as  above  explained. 

Thus  we  have  very  satisfactory  and  perfect  working  skew-bevel 
gears  of  the  kind  shown  in  Fig.  198,  with  straight  line  contacts  be- 
tween good  shaped  teeth,  though  the  gears  extended  both  ways  have 
not  contacts  far  from  the  gorge  circles,  but  are  capable  of  driving 
either  way.  These  gears  appear,  however,  like  cylindric  skew  gears, 
but  are  in  reality  skew  bevel,  as  taken  at  the  gorge  of  the  hyperbo- 


184  PRINCIPLES   OF   MECHANISM. 

loids,  though  with  no  fixed  relation  between  the  gorge  radii,  and 
angles  between  axes  and  line  of  contact. 

If  the  gears  are  at  right  angles  the  working  straight  line  contacts 
will  all  be  within  the  rectangle  abcO  for  driving- 
one  way,  and  in  the  rectangle  defO  for  driving 
the  opposite  way,  while  the  appreciable  inter- 
ference will  be  outside  the  dotted  curves  g  and 
U,  as  shown  near  0.  The  rectangles  are  de- 
termined by  the  intersections  of  the  adden- 
dum surfaces  with  the  common  tangent  plane 
FIG.  201.  between  the  two  cylinders  A  and  B,  which 

cylinders  form  the  bases  of  the  involute  teeth.     The  greatest  work- 
ing length  of  gear  A  is  therefore  ce,  and  of  B  it  is  bf. 

RESULTS  FROM  AN  EXAMPLE. 

To  better  fix  the  ideas,  a  particular  example  is  referred  to  of  the 
above  cylindric  skew-bevel  screw-gears,  in  which  the  gorge-circle 
cylinder  pitch  surfaces  extended  indefinitely  each  way,  as  shown  in 
Fig.  202,  are  5.1"  and  2.2"  in  diameter,  with  axes  at  right  angles. 
For  an  addendum  of  0.32"  the  interference  was  found  inconsidera- 
ble, but  for  thicker  addenda  appreciable  interference  would  begin 
at  about  the  dotted  lines  g  and  h,  Fig.  201,  when  gO  and  hO  equal 
about  0.8".  Then,  in  case  of  the  thicker  addenda,  with  clearance  for 
interference  cut  from  (7  up  toward  D,  and  E  up  toward  D,  Fig.  198, 
on  the  face  the  necessary  amount,  it  is  found  that  there  will  be  no 
contacts  at  or  about  0,  Fig.  201,  for  a  distance  from  0  of  about  the 
same  value  as  the  distance  from  the  dotted  curves  g  and  //  over  to 
where  the  interference  for  the  actual  addenda  ceases,  and  probably 
within  some  such  shaped  outline  as  the  curves  g  and  Ji.  These 
quantities  are  admitted  to  be  only  approximate. 

We  may  Demonstrate  the  Principle  respecting  the  above  exact 
construction  of  Olivier  by  use  of  Fig.  202 — in  some  respects  the 
same  as  one  given  by  Grant.  Here,  for  the  pitch  surfaces  of  the 
proposed  skew  gears,  we  take  AD  and  BE,  the  cylinders  of  the 
gorge  circles,  touching  at  0,  between  which  cylinders  in  common 
tangency  is  a  plane  GG,  upon  which  are  parallel  lines,  the  normal 
pitch  of  teeth  apart.  Now  suppose  the  plane  to  be  moved  along 
with  its  parallel  lines  maintained  parallel  to  a  fixed  line,  and  that 
the  cylinders  AD  and  BE  are  revolved  by  the  plane  in  such  a  way 
as  to  allow  no  lateral  slipping  of  the  parallel  lines  of  G  on  the 


AXES   CROSSING    WITHOUT   MEETING. 


185 


•cylinders,  while  at  the  same  time  admitting  endlong  sliding  of  the 
lines  in  their  own  tracks  on  the  cylinders  to  any  extent.  It  is  plain 
that  if  the  lines  could  leave  an  impression  of  themselves  on  the 
cylinders  the  latter  would  appear  like  perfect  screws,  having  a  num- 
ber of  linear  threads. 

Now  suppose  that  the  cylinders  had  been  covered  with  some 
easily  cut  material  during  this  rolling,  and  that  the  material  within 
the  surfaces  cut  by  the  lines  on  G  remained  attached  to  the  cylin- 
ders. It  is  plain  that  the  cylinders  with  these  thread  ridges  thus 


B 


FIG.  202. 

formed  and  attached  would  be  like  that  of  Fig.  198,  the  surfaces 
of  the  'threads  being  of  involute  form  in  normal  actions  and  with 
straight-lined  elements  in  all  lines  tangent  to  the  cylinders,  as  for 
FGH,  Pig.  198.  Also,  it  is  easily  seen  that  these  surfaces  are  tan- 
gent to  each  other  within  the  angles  A  OB  and  D  OE  above  and  be- 
low 0,  and  that  these  lines  of  tangency  will  all  be  in  the  plane  G, 
because  this  plane  is  normal  to  all  the  cut  surfaces.  Theoretically, 
iheo,  these  >tlirjeads,  or  spiral  ridges,  will  have  perfect  right-lined 


186  PRINCIPLES   OF    MECHANISM. 

contacts  within  the  angles  AOB  and  DOE  to  any  extent  from  0, 
and  exactly  the  form  of  contacts  required  for  gear  teeth. 

In  the  above,  no  particular  angle  has  been  assumed  between  the 
cylinders,  nor  between  either  one  of  them  and  the  cutting  lines  on 
the  plane  G.  It  appears,  then,  that  the  cylinders  may  have  any 
possible  angular  relation  with  each  other,  or  with  the  lines  on  the 
plane  G\  and  that  if  the  lines  be  assumed  as  running  from  0  into 
the  angle  AOE  instead  of  the  angle  AOB  the  contacts  will  all  be 
within  the  angles  AOE  and  BOD,  so  that  the  case  is  perfectly 
general. 

In  all  cases  there  will  be  some  degree  of  interference,  as 
explained  at  Figs.  199  and  201,  in  the  angles  opposite  those  within 
which  the  contacts  are  found. 

On  account  of  this  interference  it  is  advisable  in  practical 
gearing  of  this  kind  extending  past  0  that  the  addenda  be  com- 
paratively light — say  in  the  neighborhood  of  a  fifth  of  the  mean 
gorge  radius — and  the  teeth  correspondingly  small. 

In  these  gears,  though  perhaps  called  skew  bevels,  if  the  teeth 
extend  to  sharp  tops  throughout  they  will  be  cylindrical  screws,  as 
in  Fig.  198,  there  being  alternating  ridges  or  teeth,  and  spaces,  as. 
at  D  and  1. 

In  practical  gearing,  the  angles  between  the  lines  of  G  and  axes 
A  and  B  must  be  such  that  the  normal  pitch  will  divide  the  pitch 
cylinders  without  remainders. 

Gears  of  this  kind,  instead  of  being  taken  at  the  gorge  circles 
0,  may  be  selected  at  a  distance  from  the  gorge  circles,  as  at  A'  and 
#',  Fig.  202,  and  comparatively  short.  But  if  the  teeth  go  to  sharp 
tops  they  will  be  short,  many-threaded,  cylindric  screws.  The 
teeth  of  these  gears  will  be  comparatively  flat  and  practically 
useless  if  selected  far  from  the  gorge,  though  they  will  work  per- 
— — |  fectly  in  theory.  The  gear  A',  Fig.  202,  is 
shown  in  Fig.  203  with  skew  omitted,  where 
AJis  the  cylinder  AD  of  Fig.  202,  and  A' 
the  gear. 

The  involutes,  as  extended,  intersect  at 
various  points  M,  N,  etc.;  and  the  larger 
gears,  like  A',  require  no  bottoming  clear- 
ance, such  as  needs  to  be  provided  for  gorge- 
circle  gears,  as  at  UK,  Fig.  198. 

ft  finf1iiy  appears  that  we  have  no  exact 
theory  for  skew-bevel  teeth  that  are  of  practicable  utility,  except 


AXES    CROSSING    WITHOUT   MEETING. 


187' 


at  or  near  the  gorge  circles,  those  selected  at  a  considerable  dis- 
tance from  the  gorge  being  worthless  from  excessive  flatness  of 
teeth,  and  doing  more  crowding  than  driving.  Good  practical  skew 
bevels  must,  therefore,  come  from  the  approximate  solution  of 
Fig.  195,  or  be  fitted  with  the  assumed  and  conjugated  teeth  of 
MacCord. 

It  is  plain  that  the  velocity-ratio  changes  with  a  change  of  the 
inclination  of  the  lines  on  the  plane  G,  Fig.  202.  This  controverts 
the  statement  in  Prof.  MacCord's  Kinematics,  page  365,  respecting 
twisted  skew  wheels. 

The  Olivier  Skew  Bevels  find  application  in  such  examples  as 
Figs.  204  and  205.  To  design  those  of  Fig.  204,  draw  the  axes  AO 


FIG.  204. 


FIG.  205. 


and  BO,  and  generating  line  DO,  as  answering  to  one  of  the  lines  on 
the  plane  O,  Fig.  202.  Then  on  a  line  perpendicular  to  DO  lay  off 
the  distances  06*  and  Oc,  equal  to  the  number  of  teeth  JVTand  n  in 
the  large  and  small  gears  respectively,  multiplied  by  the  normal 
pitch.  Draw  the  triangles  OCa  and  Ocb,  right  angled  at  (7  and  c, 
and  with  Oa  and  Ob  perpendicular  to  the  axes.  Then  the  diameters 
or  radii  of  the  wheels  will  be  in  the  relation  of  the  lengths  Oa  and 
Ob. 

In  Fig.  205  one  wheel  is  a  spur  gear,  answering  to  the  case  in 
the  Olivier  gears  where  the  lines  on  plane  G,  Fig.  202,  are  parallel 
to  one  of  the  axes.  The  diagram  is  lettered  for  similar  quantities 
as  in  Fig.  204. 

To  cut  these  teeth,  so  as  to  make  the  wheels  Olivier  wheels,  the 
cutters  must  be  of  such  special  forms  aa  to  give  to  the  teeth,  in 
sections  normal  to  the  axis,  the  involute  form  outside  the  pitch 
lines.  But  this  is  hardly  'to  be  expected  in  practice,  and  ordinary- 


188  PRINCIPLES   OF   MECHANISM. 

cutters  will  be  used,  even  if  the  teeth  have  contacts  at  points 
only. 

In  Figs.  204  and  205  either  or  both  wheels  may  be  enlarged  to 
the  rack,  which  in  action  will  slide  endlong  in  their  own  tracks. 

The  Worm  and  Gear  may  be  made  as  Olivier  gears  by  making 
one  wheel  very  much  smaller  than  the  other,  and  giving  the  smaller 
one  only  one  ridge  or  screw  thread,  by  properly  inclining  the  lines 
on  plane  G,  Fig.  202,  and  by  use  of  the  proper  shaped  cutter. 

But  this,  again,  can  hardly  be  expected  in  practice,  as  good 
working  worm  wheels  are  produced  by  using  a  hob  cutter  of  the 
.same  size  and  shape  as  the  worm  itself  to  finish  cutting  the  gear. 

The  normal  pitch  should  be  the  same  here  at  the  pitch  lines 
lor  the  worm  as  for  the  gear.  When  the  axes  are  at  right  angles,  as 
usually  the  case,  the  pitch  of  the  worm  in  a  direction  parallel  to  the 
axis  is  the  same  as  the  circumferential  pitch  of  wheel  at  pitch  line. 
The  wheel  teeth,  instead  of  being  cut  straight,  are  concave,  as 
formed  by  the  revolving  hob  cutter,  thus  securing  more  bearing 
•surface  between  teeth. 

In  a  section  taken  by  a  plane  normal  to  the  wheel  axis  and  con- 
taining the  axis  of  worm,  the  shapes  of  the  teeth  of  worm  and  wheel 
•should  be  the  same  as  in  the  rack  and  pinion  of  Figs.  162  or  167, 
with  the  same  points  observed  as  to  interference  of  teeth.  The 
worm  may  be  single  or  many  threaded. 

The  Hindley  or  "Corset-shaped"  worm  may  be  supposed  to 
possess  advantages  over  the  cylindric  worm  when  examined  in  sec- 
tion; but  a  little  consideration  will  show  that  its  tooth  contacts  with 
the  mating  wheel  are  more  like  points,  or,  at  best,  a  line  on  each 
tooth  from  top  to  bottom,  because  of  the  continually  varying  diam- 
eter of  the  worm,  so  that  no  thread  can  fit  the  teeth  of  the  wheel 
-as  well  as  the  cylindric  worm  thread  can,  with  line  contacts  along 
the  thread  from  side  to  side  of  wheel;  and  second,  that  the  larger 
average  diameter  of  the  worm  will  give  rise  to  more  rapid  wear. 

Skew-bevel  Pin  Gearing  is  possible  for  all  angles  between  axes, 
but  the  pins  take  complex  shape,  except  for  the  one  case  of  right- 
angled  axes  and  velocity-ratio  unity;  when  the  pins  are  cylindrical 
and  in  diameter  equal  the  shortest  distance  between  axes. 

Worm  pin  gearing  is  also  possible;  but  as  these  pin  wheels  are 
scarcely  demanded  in  practice,  they  will  not  be  described  here. 
Detailed  descriptions  are  found  in  MacCord's  Kinematics,  pp.  284- 
293, 


AXES   CROSSING   WITHOUT   MEETING.  189* 

INTERMITTENT  MOTIONS. 

Movements  of  this  class  are  given  in  Figs.  14  to  16,  where  the 
pitch  lines  were  discussed.  The  teeth  on  these  wheels  are  the  same 
as  those  already  considered,  for  both  the  principal  circular  arcs  and 
the  non-circular  initial  arcs,  and  also  for  the  starting  and  stopping 
accessories,  as  explained  in  Figs.  140  to  143. 

These  movements  may  be  made  for  the  three  cases  of  (1)  Axes 
Parallel,  (2)  Axes  Meeting,  and  (3)  Axes  Crossing  Without  Meeting; 
but,  as  the  principles  for  all  these  have  been  duly  considered  for 
non-circular  pitch  lines,  they  will  not  be  taken  up  here  for  the 
simpler  case. 


CHAPTER  XVII. 
ALTERNATE  MOTIONS. 

I.    LIMITED  ALTERNATE  MOTIONS. 

DIRECTIONAL   RELATION    CHANGING. 

First.  With  Solid  Engaging  and  Disengaging  Parts. 

IN  Fig.  206  is  illustrated  a  movement  of  this  class  with  engaging 
:and  disengaging  features,  all  designed  with  a  view  of  being  a 
thoroughly  practical  and  durable  working  movement.  It  is  laid 
out  with  involute  teeth. 

If  revolving  right-handed,  before  the  tooth  c  can  strike  upon  e, 
the  tooth  a  will  have  moved  to  some  extent  upon  its  mate  b  and  to 


FIG.  206. 

have  thrown  the  rack  fully,  so  that  interference  cannot  occur  at  ce. 
Then,  to  start  B  gradually,  the  portion  at  d  is  carried  up  near  to 
the  center  of  A,  so  that  the  first  impulse  for  moving  B  is  received 
at  d,  where  the  velocity  of  A  is  less  than  at  «,  and  thus  reducing 
the  initial  blow.  This  may  be  carried  still  farther  toward  the  axis 
A  if  desired. 

In  this  construction  there  will  be  a  momentary  pause  of  the 
rack  at  the  end  of  the  movement. 

190 


ALTERNATE   MOTIONS. 


191 


Second.  With  Attached  Engaging  and  Disengaging  Spurs. 

In  Fig.  207  the  movement  has  attached  spurs  to  control  the  re- 
versal of  movement  so  that  the  blow  may  be  reduced.  This  is  taken 
from  a  working  design.  The  teeth  are  epicycloidal,  though  they 
may  be  of  any  form  preferred. 


FIG.  207. 

The  shock  due  to  initial  motion  is  reduced  to  a  minimum  by 
carrying  the  spurs  up  so  far  that  the  normal  to  the  curves  of  spur 
and  pin  will  strike  so  close  to  the  axis  of  A  as  to  indicate  easy 
starting. 

For  simplicity,  the  movement  of  Fig.  206  has  the  advantage, 
though  this  will  be  accompanied  with  less  shock  at  reversal.  The 
spurs  may  be  carried  still  higher  to  farther  reduce  the  shock. 

The  Mangle  Wheel. 

Under  velocity-ratio  constant  the  pitch  lines  must  be  circular, 
and  in  this  movement  the  wheel  is  limited  to  one  revolution  at 
most,  an  example  of  which  is  shown  in  Fig.  104. 

In  this  form  of  axes  meeting  the  wheel  may  have  ordinary  teeth 
on  the  edges  of  a  band  instead  of  pins;  but  for  a  face  wheel  pins 
would  be  necessary,  otherwise  the  opposite  sides  of  a  band  bearing 
teeth  would  give  different  radii  of  pitch  lines,  and  hence  different 
velocities  for  forward  and  return  motion.  This  construction  may, 
however,  be  required,  giving  a  case  of  velocity-ratio  changing, 
though  constant  for  each  direction. 

II.    UNLIMITED  ALTERNATE  MOTIONS. 

These  are  found  in  the  mangle  rack,  which  may  be  conceived 
of  by  supposing  the  wheel  of  Fig.  104  to  be  cut  at  the  center  of  the 
H-shaped  reversing  piece,  then  straightened  out  to  a  rack,  and 
then  mounted  on  proper  slides. 

The  teeth  may  be  on  the  upper  and  lower  sides  of  a  rack  bar 
instead  of  a  row  of  pins.  Any  length  may  be  given  to  the  move- 
ment, and  hence  it  may  be  classed  as  unlimited. 


CHAPTER  XVIII. 
.     CAM   MOVEMENTS. 

IN  a  cam  movement  the  path  and  the  law  of  motion  of  the  fol- 
lower may  be  determined  independently  of  the  driver,  when  the 
latter  is  made  to  conform  therewith. 

Thus  the  follower  path  may  be  a  straight  line,  a  circle  or  any 
other  curve,  and  the  movement  of  the  follower  in  that  path  may  be 
assumed  point  by  point  for  the  full  forward  and  also  for  the  back- 
ward movement. 

The  cam  is  accompanied  by  an  undue  amount  of  friction  by  the 
rubbing  of  the  follower  on  the  driver,  which  in  turn  contributes  to 
the  wear  of  the  parts,  causing  backlash  and,  in  high  speeds,  noise. 

This  movement  is  the  one  usually  called  to  his  aid  as  the  last 
resort,  when  the  designer,  seeking  to  avoid  it,  'fails  to  obtain  the- 
necessary  motion  of  a  piece  by  other  means,  such  as  may  be  pro- 
posed for  lighter  and  more  quiet  running  of  parts. 

The  cam  is,  therefore,  a  most  useful  movement,  and  in  certain 
cases  must  be  accepted,  though  to  be  avoided  whenever  possible. 

CAMS  IN  GENERAL. 

BY  CO-ORDINATES. 

The  driver  may  have  a  non-uniform  motion  as  well  as  the  fol- 
lower. To  unite  all  questions  in  a  general 
solution  of  a  cam  and  follower  in  one 
plane,  let  A,  Fig.  208,  represent  the  driver 
axis,  to  which  is  attached  a  pointer,  A\, 
the  same  turning  so  as  to  be  in  positions 
2,  3,  4,  etc.,  at  the  ends  of  equal  successive 
intervals  of  time.  The  unequal  angles  at 
A  indicate  variable  motion  of  A.  Let  the 
curve  1,  2,  3,  etc.,  at  D  represent  the  path 
of  the  follower,  D  the  point  of  the  latter,. 
which  reacts  against  the  cam,  being  at  positions  2,  3,  4,  etc.,  at  the 
ends  of  equal  successive  intervals  of  time,  these  intervals  being  the* 

192 


CAM    MOVEMENTS. 


193 


same  as  for  A.     The  mechanism  by  which  D  is  mounted,  compel- 
ling it  to  move  in- the  path  1,  2,  3,  etc.,  is  not  shown. 

Draw  a  circle  arc  la  from  the  follower  path  to  the  position  1  of 
the  driver  pointer.  Likewise  arcs  2b, 

3c,  4d,  etc.,  as  shown.     This  gives  us     /^^     ^^    X  7\/D 
co-ordinates  by  which  to  construct  the 
cam,  as  m  Fig.  209,  where  «1,  J2,  c3, 
etc.,  are  laid  off  from  the  initial  line 
Al. 

Then  the  cam  outline  may  be  drawn  FIG.  209. 

in  as  the  curve  1,  2,  3,  4,  etc.     In  practice  this  may  all  be  done  in 
the  same  figure. 

This  cam,  .45,  while  turned  from  the  position  shown  around  till 
5  comes  to  E,  will  evidently  drive  the  follower's  reacting  point 
along  in  its  path  from  1  to  E.  The  arc  E5,  from  the  construction, 
evidently  equals  the  arc  \e  of  Fig.  208,  as  it  should,  and  likewise 
for  the  other  arcs.  Hence  the  follower  will  move  in  its  path  as 
proposed,  while  A  has  the  motion  assigned  to  it. 

BY  INTERSECTIONS. 

The  cam  may  be  drawn  by  the  method  of  intersections,  as  iir 

Fig.  210,  where  the  follower  path  is 
[D  laid  off  in  the  several  positions  in  the 
inverse  order  of  motion  of  A,  following 
the  successive  equal  intervals  of  time, 
as  shown. 

For  convenience  in  this,  a  templet 
may  be  cut  to  the  follower  path  curve, 
FIG.  210  with  center  point  at  A  noted.     Then 

with  the   angles   1^42,  2^43,  etc.,  laid  off  and  noted,  the  several 
curves  can  be  struck  by  templet. 

Now,  drawing  in  the  circle  arcs  from  the  points  1,  2,  3,  etc.,  of 
the  follower  path,  we  obtain  intersections  and  can  draw  the  curve 
1,  2,  3,  4,  etc. 

The  method  of  intersections  is  usually  employed  in  practice, 
and  as  the  motion  of  the  driver  A  is  generally  uniform,  the  path 
positions  1,  2  2,  3  3,  etc.,  are  uniformly  distributed. 

DIRECTIONAL  RELATION  AND  VELOCITY-RATIO. 

In  the  above  cases  the  directional  relation  is  constant  but  for 
the  follower  to  return  to  its  starting  point,  as  A  continues,  it  must. 


194  PRINCIPLES   OF   MECHANISM. 

be  changing.  For  this  we  may  draw  the  cam  in  the  same  way,  lay- 
ing off  another  set  of  points  in  its  path  for  the  return  of  the  fol- 
lower, and  the  corresponding  angles  of  A. 

When  the  angles  for  A  are  equal,  and  the  points  1,  2,  3,  etc., 
equidistant,  the  velocity-ratio  is  evidently  constant,  regarding  the 
follower  as  that  which  moves  in  the  path  1,  2,  3,  etc. ;  otherwise 
not.  Sometimes  the  follower  moves  in  a  straight  line,  and  some- 
times swings  about  a  center  B,  the  velocity  of  which  in  either  case 
is  to  be  compared  with  A. 

It  seems  unnecessary  to  classify  cams  under  directional  relation 
and  velocity-ratio,  and  they  will  be  treated  here  without  it. 

VELOCITY-RATIO  FOE  A  SWINGING  FOLLOWER. 

In  Fig.  211  we  have  a  cam  A,  and  a  follower  BD,  which  swings 
about  an  axis  B.  The  path  of  the  point  D  which  acts  against 


FIG.  211. 

the  cam  is  a  circle  arc,  EF.  To  find  the  velocity-ratio  between  A 
and  B,  draw  the  normal  DC,  and  then,  according  to  Figs.  105  and 
106,  the 

ang.  velocity  of  B      AC 

velocity -ratio  =  - ; — r^ — j— -.  =  -^-^, 

ang.  velocity  of  A      £C 

being  thus  in  the  inverse  ratio  of  the  segment  of  the  line  of  centers, 
as  has  been  found  for  other  movements. 

When  the  highest  point  of  the  cam  passes  under  D,  the  normal 
CD  will  strike  at  A  and  the  angular  velocity  of  B  will  be  zero. 
As  A  moves  on,  the  point  C  changes  to  the  other  side  of  A,  and  the 
motion  of  B  will  be  reversed,  thus  changing  the  directional  relation. 
When  C  is  between  A  and  B  the  directions  of  motion  are.  opposite, 
while  for  C  outside  the  directions  are  alike. 


CAM   MOVEMENTS. 


195 


VELOCITY-RATIO  WHEN  THE  FOLLOWER  MOVES  IN  A  STRAIGHT  LINE. 

When  the  follower  point  D  moves  in  a  straight  line,  B  in  effect 
is  at  an  infinite  distance  away,  as  in  Fig.  212,  and  BD  =  B C  —  in- 
finity.    Then  the  linear  velocity  at  D  is  to  be  com- 
pared with  the  angular  velocity  of  A  for  velocity- 
ratio. 

To  find  the  linear  velocity  of  D,  we  have  from 
the  above  the 

linear  velocity  of  D  —  BO  X  ang.  velocity  of  By 
and  this  velocity-ratio,  Fig.  212,  will  be 
linear  velocity  of  D  _       ~ 
ang.  velocity  of  A 

To  draw  an  interpretation  from  this,  take  the 
expression  in  the  form 

linear  velocity  of  D  =  AC  X  ang.  velocity  of  A. 
This  signifies  that  the  linear  velocity  of  D  in  its 
straight  path,  and  for  the  position  considered,  is 
equal  to  the  velocity  of  the  point  0  as  if  it  were 
fixed  upon  the  cam  A  and  revolving  with  it. 


FIG.  212. 


CONTINUOUSLY  REVOLVING  CAM,  AND  RETURNING 
FOLLOWER. 

BY  METHOD  OF  INTERSECTION. 

Assume  the  motion  of  A  uniform,  and  that  the  follower  swings 
about  an  axis  B,  Fig.  213,  causing  D  to  move  in  a  circular  path, 
D,  1,  2,  3,  4,  forward,  and  5,  6,  7,  8,  D  on  the  return.  Draw  circles 
through  these  points.  There  being  nine  divisions  in  the  follower 
path,  make  also  nine  corresponding  equal  divisions  in  a  circle 
struck  through  B.  From  each  point  of  division  as  a  center  and 
with  a  radius  BD  strike  the  nine  circle-arc  follower-path  lines  1, 
2,  3,  etc.,  as  shown,  intersecting  the  parallel  circular  lines  drawn 
on  the  cam  from  A. 

Then  draw  in  the  linear  cam  outline  through  the  points  of  in- 
tersection of  these  lines  of  like  number. 

Numbers  in  the  circles  about  A  are  in  order  left-handed. 
Therefore  the  cam  is  to  revolve  in  the  inverse  order,  right-handed. 

Here  the  follower  moves  forward  and  then  back  to  the  point  of 
beginning  in  one  revolution  of  A,  and  the  movements  may  be  re- 


196 


PRINCIPLES    OF    MECHANISM. 


peated  indefinitely.  The  forward  movement  of  D  is  faster  than 
the  return,  there  being  four  and  five  divisions  to  each  respectively. 
These  divisions  of  path  are  to  represent  the  velocity  of  D. 


FIG.  213. 

Here  the  directional  relation  is  changing  and  the  velocity-ratio 
varying. 

CASE  OF  A  CYLINDRICAL  GAM. 

Here  A  A  is  the  elevation  of  the  cylindric  cam  and  0  the  plan 
showing  eight  equal  divisions,  as  in  Fig.  214.     At  EF  is  the  de- 


>4 


D 


008 


FIG.  214. 


velopment  of  the  cylinder,  showing  the  linear  cam,  the  same  being 
also  shown  on  the  cylinder. 

D  is  the  point  of  the  follower  moving  in  a  straight  path  parallel 
to  the  cylinder,  so  that  the  follower-path  lines  in  the  development 


€AM    MOVEMENTS. 


197 


are  straight  and  vertical.  Above  E  are  laid  off  the  points  1, 2,  3,  4, 
etc.,  in  the  follower  path,  and  through  them  are  drawn  the  parallel 
cam  lines.  Then  the  linear  cam  line  is  to  be  drawn  through  inter- 
sections  of  lines  of  like  number,  as  shown. 

By  redeveloping  upon  the  cylinder,  we  obtain  the  cylindric 
linear  cam.  In  practice,  the  drawing  may  be  made  upon  the  cylin- 
der itself  direct. 

The  case  of  the  figure  is  a  simple  one  of  a  right-line  movement 
of  the  follower. 

The  double  screw  movement  is  an  example  of  a  double  cam  mo- 
tion, where  a  reciprocation  is  effected  by  several  turns  of  the  screw 
cam,  the  threads  or  cam  grooves  returning  and  crossing  themselves. 
The  follower  in  this  case  is  oblong,  and  of  length  sufficient  to  guide 
it  in  one  groove  as  it  passes  another. 

In  case  of  a  circular  path  for  D,  the  vertical  or  follower-path 


/  I  I  ! 


FIG.  215. 

lines  in  the  development  would  be  circular  and  copied  from  the 
path,  as  shown  in  Fig.  215. 

It  is  possible  for  the  path  of  D  to  be  some  other  curve,  in  which 
case  the  path  lines  on  EF  should  be  copied  from  it. 


CASE  OF  A  CONICAL  GAM. 

This  is  similar  to  Fig.  214,  the  parallel  cam  lines  around  the 
conic  surface  developing  in  circles  as  in  Fig.  216.  The  follower 
path  is  here  taken  as  an  element  of  the  cone  for  a  simple  case. 
But  the  path  of  D  may  be  a  circle  arc  or  other  line  when  the  fol- 
lower-path lines  of  EF  will  of  course  be  drawn  to  the  same  circle 
arc  or  other  curve  as  that  of  the  path. 


198 


PRINCIPLES   OF   MECHANISM. 


CASE  OF  A  SPHERICAL  CAM. 

Here  the  sphere  A  A  is  to  serve  for  the  cam,  as  well  adapted 
for  the  case  of  axes  A  and  B  at  right  angles  and  intersecting,  the 


FIG.  216. 
sphere  being  centered  at  the  point  of  intersection,  as  in  Fig.  217. 

rn 


FIG.  217. 
The  position  of  the  follower  point  D  is  so  chosen  that  the  angle 


CAM    MOVEMENTS.  199 

D" A" B"  is  a  right  angle,  and  the  motion  of  D  in  its  path  will 
describe  a  meridian  to  the  sphere.  Then  the  proper  number  of 
meridians  and  parallels  are  to  be  drawn  for  path  lines  and  cam 
parallels,  the  latter  through  the  points  1,  2,  3,  etc.,  of  the  path,  as 
seen  in  plan  at  A'D'A'.  The  linear  cam  may  now  be  drawn  as 
before,  through  intersections  of  lines  of  like  numbers. 

It  is  easy  to  imagine  a  less  simple  case,  as,  for  instance,  where 
the  angle  BDA  is  not  a  right  angle  for  which  a  follower-path  line 
of  D  on  the  sphere  would  not  be  on  a  meridian,  but  on  an  oblique 
great  circle. 

Also,  the  line  of  B  might  be  at  a  considerable  distance  back  or 
in  front  of  the  axis  AA,  as  for  the  case  of  axes  crossing  but  not 
meeting,  when  the  cam  blank  ADA  will  be  in  the  form  of  a  circu- 
lar spindle,  or  a  circular  zone,  respectively. 

A  PLANE  CAM  OR  CAM  PLATE. 

For  this  the  cam  will  be  like  a  sliding  plate,  a  cam  plate;  as  if 
the  development  of  Fig.  214  or  215  were  mounted  to  slide  on 
guides  in  reciprocation,  or  the  corresponding  part  of  Fig.  216  to 
swing  about  a  pivot. 

CAM  WITH  FLAT-FOOTED  FOLLOWER. 

In  some  cam  movements  the  follower  has  a  flat  bearing  piece, 
D,  Fig.  218,  instead  of  a  point,  which  for  the  same  cam  changes 
the  law  of  motion  of  the  follower,  but  it  presents  a  more  extended 
bearing  surface  to  the  cam.  It  is  evidently  immaterial  where  the 
guide  rod  B  is,  whether  to  right  or  left,  provided  the  pressure  at 
D  does  not  cramp  B  excessively.  Again,  D  may  be  mounted  to- 
swing  about  a  center,  Bf ',  as  shown  in  dotted  lines. 

The  velocity-ratio  for  B  is  to  be  found  as  before,  with  respect  to 
the  normal  DC. 

CAMS  FOR  SPECIFIC  LAW  OF  MOTION  or  FOLLOWER. 

1st.  A  Uniform  Motion  of  Follower,  in  Reciprocation  in  a 
Straight  Line. 

In  Fig.  219,  take  D,  1,  2,  3,  4  as  equidistant  points  in  a  straight 
follower  path  which  passes  the  axis  A  at  a  distance  EA  from  it, 
Draw  parallel  cam  circles  through  the  points  0  to  4,  and  the  fol- 


200 


PRINCIPLES   OF    MECHANISM. 


lower-path  lines  as  tangent  to  the  circle  EA,  cutting  the  outer 
oircle  at  equidistant  points,  1,  2,  3,  4,  etc.,  to  8. 


FIG.  218. 


FIG.  219. 


Then  the  linear  cam  can  be  traced  in  as  shown,  which,  as  a 
cam  acting  on  D,  will  impart  to  it  a  uniform  motion,  0  to  4  and  4 
to  0,  with  the  same  constant  velocity-ratio. 

The  curve  DUG  will  be  an  involute  to  the  circle  EA  when  the 
length  of  path  is  equal  the  half  circumference  of  EA,  and  is  the 
correct  cam  curve  for  the  lifting  cams  of  a  miner's  stamp  mill,  the 
stamp  head  rods  being  in  the  line  ED. 

The  velocity-ratio  being  constant,  the  normals  to  the  linear  cam 
curve  DUG,  at  any  contact  of  D,  will  all  intersect  the  line  EA 
produced  in  one  and  the  same  point  E  at  the  left  of  A.  Also,  the 
normals  to  the  cam  curve  DIG  will  intersect  the  line  EA  at  the 
same  distance  from  A  on  the  opposite  side. 

If  the  construction  be  such  that  the  point  E  coincides  with  A, 
the  linear  cam  curves  become  Archimedean  spirals,  and  the  cam 
one  of  constant  diameter,  and  known  as  the  heart  cam. 


2d.  Law  of  Motion  that  of  the  Crank  and  Pitman. 

In  Fig.  220,  A  is  the  center  of  the  cam,  0  the  center  of  the 
shaft,  and  abcde  half  orbit  of  the  crank  pin,  in  which  the 


CAM   MOVEMENTS. 


201 


latter  is  supposed  to  move  at  a  uniform  rate,  making  the  spaces  ab, 
•be,  etc.,  in  equal  times. 

The  pitman  has  the  length  ao,  ID,  etc.,  e4,  so  that  the  spaces  0  1, 
1  2,  etc.,  are  passed  in  equal  times  by  the  end  D  of  the  pitman  bD, 
and  with  a  varying  movement  in  accordance  with  the  so-called 
•crank  and  pitman  motion.  Through  these  points  draw  the  par- 


FIG.  220. 

;allel  cam  circles  and  divide  them  into  equal  parts  by  the  follower- 
path  lines  as  tangents  to  the  circle  A  C  and  to  equal  points  of 
'division,  0,  1,  2,  3,  etc. 

Then  draw  in  the  linear  cam,  through  points  of  intersection  of 
path  and  cam  circle  lines  of  like  numbers.  This  cam  will  give  the 
follower  D  the  same  motion  as  would  the  crank  and  pitman;  the 
crank  or  the  cam  both  being  supposed  to  revolve  with  uniform 
motion. 

3d.  Law  of  Motion  that  of  a  Falling  Body. 

In  Fig.  221,  A  is  the  center  of  the  cam,  D  the  follower  to  move 
in  the  path  /)4,  its  velocity  in  the  first  half  of  which  path  is  to  be 
accelerated  as  that  of  a  falling  body,  and  in  the  last  half  to  be 
retarded  according  to  the  same  law. 

To  express  this  law  on  the  line  Z>4  of  the  path,  draw  a  parallel 
-ag,  on  which  as  an  axis  draw  a  parabola  abc,  and  an  equal  one, 
gdc,  intersecting  it  at  c.  Make  several  equal  divisions  gf,fe,  etc., 
•and  from  these  point  off  equal  divisions,  erect  ordinates  ec,fdb,  etc., 
and  from  the  points  of  intersections,  d,  c,  b,  etc.,  draw  horizontal 
lines  over  to  the  path  />4.  Then  if  the  spaces  nj,ji,  etc.,  or  0  1, 


202 


PRINCIPLES    OF    MECHANISM. 


1  2,  etc.,  down  to  the  middle  of  ag  are  described  in  equal  timesr 

the  law  of  motion  is  that  of  a  fall- 
ing body,  from  the  known  prop- 
erty of  the  parabola.  The  re- 
versed parabola  gdc  will  give  the 
desired. points  in  retardation. 

One  peculiar  property  of  the 
action  here  is  that  the  inertia 
of  the  parts  of  the  follower  will 
cause  a  constant  resistance  in  the 
follower  path — a  probable  advan- 
tage where  the  follower  is  very 
heavy. 

In  practice,  numerous  lines 
should  be  used  in  the  construc- 
tion, a  few  only  being  employed 
FIG.  221.  here  for  clearness  of  figure. 

4th.  Tarrying  Points  for  Follower. 

It  often  happens  that  the  follower  is  to  make  a  movement,  then 
halt  momentarily,  then  make  another  movement,  then  halt, 
These  halting  or  tarrying  points  are  easily  pro- 
vided   for   in   the   cam  by  introducing   circular 
segments,  as  shown  in  Fig.  223. 

5th.  Uniform  Reciprocating  Motion. 

The  so-called  heart  cam  is  a  common  device 
for  this,  an  example  of  which  is  shown  in  Fig. 
222.  As  a  linear  cam  it  has  a  constant  diameter 
through  the  axis,  and  when  the  follower  is  located 
to  coincide  with  this  axis  and  is  provided  with  a 
bearing  point  against  the  cam,  both  above  and  below  it,  there 
will  be  no  backlash  between  the  cam  and  follower. 


Fro.  222. 


6th.  The  Heddle  Cam. 

This  can  be  used  to  work  the  heddles  or  harness  in  looms  where 
the  intervals  of  motion  and  rest  are  about  equal.  Fig.  223  illustrates, 
by  photo-process  copy,  a  working  cam  of  this  kind. 


CAM    MOVEMENTS. 


203- , 


To  make  a  heddle  cam  of  constant  diameter,  uniform  motion,, 
and  with  tarrying  intervals  of  time  equaling  the  motion  intervals : 

In  Fig.  224  divide  the  cam  into  four  equal,  parts,  retaining  the 
opposite  segments  DH  and  EG  for  the  tarrying  intervals.  Tha 
remaining  opposite  segments  must 
serve  for  the  linear  cam  lines.  In 
these  segments  draw  equidistant  cir- 
cle arcs,  and  also  radial  lines,  in  equal ; 
number.  The  linear  cam  lines  may 
then  be  drawn  in  through  points  of 
intersection  as  shown,  giving  us  the 
linear  cam  outline  DbEGeHD. 


FIG.  223. 


FIG.  224. 


All  lines  of  this  cam  through  A  will  have  the  same  length,  or 
the  cam  will  have  a  constant  diameter,  and  it  will  just  fill  the  space 
between  follower  points  D,  G  on  a  slide  BB'. 

The  curves  DIE  and  GeH  are  Archimedean  spirals. 


7th.  Easements  on  Cams. 

The  cam  of  Fig.  224  will  instantaneously  start  the  follower  into 
full  motion,  the  latter  remaining  uniform  until  stopped  with  equal 
abruptness.  This  is  objectionable  in  any  case,  and  when  the  fol- 
lower is  heavy,  it  is  destructive.  This  is  obviated  by  what  may  be 
termed  easement  curves  for  the  cam,  or  easements. 

An  Arbitrary  Easement  may  be  drawn,  as  in  Fig.  225,  where  A 
is  the  center  of  the  cam,  and  G  a  point  of  abrupt  change  between 
the  tarrying  and  cam  arcs. 

Assume  equal  angles  EAG  and  GAF,  and  divide  them   into 


204: 


PRINCIPLES   OF   MECHANISM. 


-equal  parts  by  lines  extended  outward.  Then  note  points  a,  b,  c, 
etc.,  over  to  the  line  through  G,  and  in 
increasing  distances  from  EG,  such  as 
estimated  to  give  a  proper  easement  curve 
when  drawn  through  the  points.  Lay  off 
the  distances  fa,  gb,  etc.,  on  ie,  hd,  etc., 
until  the  center  radius  Gc  is  reached. 
Then  draw  the  easement  curve  EabcdeF 
through  the  several  points. 

Treating  the  cam,  Fig.  224,  in  this  way, 
filling  in  at  E  and  G,  and  taking  off  like 
amounts,   on   like   radii  and  angles   at  D 
FIG.  225.  an(j  H,  we  modify  the  cam,  Fig.  224,  by 

supplying  easements,  and  without  altering  the  constancy  of  its 
various  diameters. 

As  thus  treated,  however,  we  encroach  upon  the  tarrying  arcs 
and  shorten  the  times  of  rest  of  the  follower.  We  may,  however, 
provide  for  the  angle  EG  in  laying  out  the  cam  angles. 

8th.  To  Confine  the  Easements  within  the  Assigned  Cam  Sectors, 
take  DAH,  Fig.  226,  as  the  allotted  sector  for  the  linear  cam  arc 
sought,  including  the  easements.  Let  it  be  assumed  that  the  ease- 
ments shall  start  the  follower  as  by  the  law  of  the  crank  and  pitman 
motion;  then,  when  full  motion  is  at- 
tained,  to  continue  that  motion  of  fol- 
lower uniform  till  approaching  the 
.arc  of  rest,  and  then  to  stop  the  fol- 
lower by  the  same  law  of  the  crank. 

To  this  end  draw  the  quarter  cir- 
cles Dl  and  GF  from  centers  K  and  / 
on  the  follower  path,  and  connect  G 
and  I  by  a  common  tangent.  Then 
with  dividers  note  points  in  equal  di- 
vision from  D  through  /and  G  to  P. 
Project  these  points  to  the  path  line 
DF,  and  draw  circle  arcs  about  A  FIG.  226. 

through  the  sector.  Also  draw  an  equal  number  of  radial  lines 
equidistant  between  D  and  H.  The  linear  cam  line  DLE  can  now 
be  drawn  in,  which  will  be  found  to  have  easements  at  D  and  E 
which  are  tangent  to  the  arcs  of  rest  for  follower,  and  without  en- 
croachment thereon. 

This  linear  cam  will  be  found  steeper  at  L  than  'the  cam  of  Fig. 


H 


CAM   MOVEMENTS. 


205- 


224,  and  it  may  be  a  question  which  cam  is  preferable.  But  by 
making  the  circle  arcs  DI  and  GF  smaller  the  easements  may  be 
more  limited,  and  the  inclination  of  the  central  portion  of  the  cam 
arc  less  severe. 

Other  laws  of  easement  curve  may  be  adopted,  as,  for  instance, 
DF  may  be  taken  as  the  base  of  a  cycloid,  in  which  the  same  num- 
ber of  divisions  may  be  made  as  in  DIGF.  If  that  arc  be  shorter 
than  the  one  here  drawn,  the  resulting  linear  cam  will  have  a  milder 
declivity  at  L. 

The  use  of  any  curve  DIGF,  which  is  symmetrical  with  respect 
to  a  perpendicular  to  the  middle  point  of  DF,  will  give  a  linear 
cam  line  DLE,  which,  if  paired  with  a  like  one  in  the  opposite 
quadrant,  as  in  Fig.  224,  will  constitute  a  cam  of  constant  diameter 
through  A. 

In  Fig.  226  the  law  of  motion  of  follower  is  represented  by  the 
spacing  along  the  follower  path  DF. 

CASE  OF  FLAT-FOOTED  FOLLOWERS. 

Let  the  law  be  that  of  a  crank  and  pitman  motion;  determine  a 
cam  of  the  kind  shown  in  Fig.  218.  Draw  the  circle  of  the  crank. 


FIG.  227. 


FIG.  228. 


at  DIE,  Fig.  227,  and  project  points  a,  I,  c,  etc.,  of  division  into 
equal  crank  arcs,  to  the  radial  line  DE,  giving  points  d,  e,f,  etc.. 


OF   THE 

UNIVERSITY 


PA  i 


206 


PRINCIPLES   OF   MECHANISM. 


Draw  the  same  number  of  diametric  lines  through  A  as  FA, 
GAy  etc.,  to  represent  the  motion  of  A.  Then  draw  cam  parallel 
circles  from  points  d,  e,  /,  etc.  At  intersections,  as  at  F,  G,  etc.,  draw 
the  follower  line  of  DM  as  representing  the  position  of  DM  relative 
to  A  when  FA,  GA,  etc.,  have  revolved  to  the  line  AD.  That  is, 
if  DM  is  perpendicular  to  AD,  draw  FJ,  GL,  etc.,  perpendicular 
to  the  lines  AF,  AG,  etc. 

Tangent  to  all  these  position  lines  of  MD  draw  in  the  linear  cam 
curve,  as  HLJD.  For  this  case  the  half  may  be  copied  on  the  other 
side  of  HAD  as  dotted,  giving  the  same  law  of  return  of  follower. 

For  this  particular  case  and  law,  the  cam  is  a  circle  or  crank  pin. 

In  cams  with  a  follower  consisting  of  a  straight  footpiece  D, 
where  the  law  on  the  return  movement  is  the  same  as  for  the  for- 
ward, the  cam  will  be  one  of  constant  breadth  as  reckoned  on  a 
normal  to  the  linear  cam;  and  the  follower  may  be  made  as  a  rack 
or  frame  DE,  Fig.  228,  with  parallel  sides. 

Thus  cams  which  while  revolving  just  fill  the  space  between  a 
pair  of  points  fixed  upon  the  follower,  as  in  Figs.  222  to  224,  and 
229,  may  be  called  cams  of  constant  diameter;  while  those  likewise 
just  filling  the  space  between  parallel  lines,  or  footpieces  fixed 

upon  the  follower,  as  in  Fig.  228, 
may  be  called  earns  of  constant  breadth. 

CAMS  OF  CONSTANT  DIAMETER  AND 
BREADTH. 

1st.  Constant  Diameter. 

Besides  those  mentioned  above, 
Figs.  222,  223,  224,  225,  and  226,  there 
may  be  cases  where  a  specific  law  of 
motion  is  to  be  followed  for  a  half 
revolution  of  cam,  while  the  motion 
for  the  remaining  half  is  immaterial 
as  to  law. 

For  this,  as  before,  the  follower 
points  DK,  Fig.  229,  must  move  in  a 
straight  line  through  the  center  A. 

Taking  DEGIK,  Fig.  229,  as  the 
essential  cam  for  the  half  revolution, 

draw  a  series  of  lines,  EF,  GH,  IJ,  etc.,  through  A,  and  make  them 
all  equal  in  length  to  DK,  noting  points  FHJ,  etc.  Then  the 
required  counter  half  of  the  cam  will  be  the  dotted  line  KFHJD. 


PIG.  229. 


CAM   MOVEMENTS. 


207 


2d.  Cams  of  Constant  Breadth  between  parallel  lines  are  possible, 
as  in  Fig.  230,  where  a  half  of  A  is  assumed,  and  the  forked 


FIG.  230. 

piece  B  to  swing  around  the  axis  will  just  span  the  half  cam 
at  DE.  Then  by  placing  the  fork  in  various  positions  as  at  F 
and  G  and  drawing  a  line  at  G,  again  at  H  and  /  and  drawing  a 
line  at  /,  etc.,  for  a  sufficient  number  of  positions,  and  drawing  an 
enveloping  curve  EGID,  we  have  the  remaining  half  of  a  cam,  the 
breadth  of  which  for  all  positions  just  fills  the  fork-shaped  follower. 

The  outline  of  the  cam,  also  its  law  of  motion  correspondingly, 
are  limited,  however,  by  the  circumstance  that  the  center  of  cur- 
vature of  every  part  of  the  outline  must  be  within  that  outline. 

3d.  A  Cam  of  Constant  Breadth  as  between  parallel  lines  may 
be  drawn  by  aid  of  a  series  of  intersecting  lines,  as  in  Fig.  231. 

Draw  any  system  of  intersecting  straight  lines.  Then,  beginning 
at  some  remote  intersection  as  at  «,  draw  a  circle  arc  A  from  line 
to  line,  as  those  intersecting  at  a.  This  arc  will  be  normal  to  the 
lines  limiting  it.  Next,  seeing  that  the 
arc  B  will  meet  the  line  intersecting  at 
b,  take  b  as  a  center  and  draw  the  arc  B. 
The  next  arc  will  be  C  drawn  from  the 
center  c,  so  that  C  will  be  normal  to 
both  lines  intersecting  at  c.  Draw  the 
arc  D  from  the  center  d\  the  arc  E  from 
e\  the  arc  F  from/;  the  arc  G  from  g\ 
the  arc  £f  from  h-,  thus  returning  to  the 
place  of  beginning.  Arc  H  closes  upon 
arc  A  exactly,  if  the  work  is  right.  Fl°-  231- 

A  little  consideration  will  show  that  this  cam,  for  every  position, 
will  just  fill  between  a  pair  of  parallel  lines  at  a  distance  A  E  apart. 


208 


PRINCIPLES  OF   MECHANISM. 


Also,  it  will  just  revolve,  with  a  close  fit,  in  a  square  hole  of  width 
AE  each  way;  or  in  a  rhomboidal  hole  of  width  AE  each  way,  and 
with  any  angle  between  the  parallel  sides.  Also,  the  hole  may  be 
bounded  by  parallel  circle  arcs,  instead  of  parallel  straight  lines, 
partly  or  entirely,  as  in  Fig.  237. 

CAMS  WITH  SEVERAL  FOLLOWERS. 

1st.  One  Cam  may  have  Two  or  More  Followers,  as  shown  in  Fig. 
232,  the  velocity-ratio  of  which  may  or  may  not  be  the  same,  as 
depending  upon  the  construction  of  followers. 


FIG.  232. 


2d.  The  Effect  of  Two  Followers  may  be  obtained  by  combining 
two  followers  into  one,  as  in  Fig.  233,  where  the  combined  parts 
DG  are  solid,  with  a  slide  branch,  E,  the  latter  working  in  a  box  on 


FIG.  233. 

a  second  sliding  piece,  F,  so  that  motion  may  be  taken  off  at  B  and 
J  in  two  directions.. 


CAM   MOVEMENTS. 


209 


3d.  Fig.  234  gives  a  Similar  Construction,  in  which  DG  swings 
about  a  pivot  E,  and  that  in  turn  about  a  second  but  fixed  pivot  F. 

The  follower  connectors  B  and  J  may  take  the  motion  in  two 
directions.  One  spring  may  serve  to  return  the  compound  follower 
for  the  two  directions  of  movement. 


FIG.  234. 

4th.  Cams  of  Constant  Breadth  may  return  the  compound  fol- 
lower positively,  as  in  Fig.  235,  where,  as  stated  at  Fig.  231,  the 
cam  of  constant  breadth  will  just  fit  and  revolve  in  a  square  opening. 


FIG.  235. 


5th.  The  Four-motion  Cam  is  shown  in  Fig.  236,  and  constructed 
by  aid  of  intersecting  lines  in  the  manner  shown  in  Fig.  231. 

When  the  lines  AFand.  AE  are  at  right  angles  and  EF  inter- 
sects them  at  equal  angles  of  45  degrees,  and  the  cam  is  drawn  from 
centers  AEF,  as  shown,  the  follower  Dy  when  pivoted  at  a  rela- 


210 


PRINCIPLES   OF   MECHANISM. 


tively  great  distance  to  the  right,  will  have  every  point  moving  in 
an  exact  square;  as,  for  example,  the  point  d  will  describe  the 


FIG.  236. 

square  defy,  where  de  equals  the  difference  of  the  construction 
radii  a  and  b. 

This  cam  has  been  used  for  sewing-machine  feeds,  for  which  it 
is  well  adapted,  except  that  for  this,  one  side  of  the  follower  should 
be  cut  out  and  an  adjustable  stop  put  to  it  to  vary  the  feed. 

In  drawing  the  cam,  it  is  immaterial  as  to  motion  how  large  it 
is,  provided  the  periphery  goes  through  or  outside  the  points  E  and 
f\  outside  being  preferable  to  avoid  sharp  intersections  at  those 
points. 

In  Fig.  237  is  illustrated  a  working  cam  of  this  kind,  where  two 


FIG.  237. 

of  the  parallel  lines  of  the  follower  are  straight,  and  two  are  par- 
allel circle  arcs  struck  from  the  upper  joint  pin.     This  cam  fits 


i 


CAM    MOVEMENTS. 


211 


closely  both  ways  in  the  opening,  and  the  circle  arc  sides  have  the 
effect  to  keep  the  vertical  connec- 
tor bar  quiet  for  two  of  the  four 
movements,  its  motion  being  inter- 
mittent in  reciprocation. 

6th.  The  Peculiar  Shaped  Cam, 
called  a  Duangle  by  Reuleaux,  is 
shown  in  Fig.  238.  The  duangle 
closely  fits  in  the  triangular  open- 
ing in  the  follower  for  the  entire 
revolution. 

The  follower  moves  forward  and 
immediately  back,  and  there  tarries 
nearly  stationary  for  about  a  sixth 
of  a  revolution,  when  another  reciprocation  is  commenced. 

An  interesting  double  cam  motion,  which  Willis  would  class  as 
resulting  in  an  aggregate  path,  is  given  in  Fig.  239.  Two  arms 
work  agreeably  to  the  end  of  carrying  a  pencil  point  in  such  a 
manner  that  it  writes  the  letters  "  O.S  U."  and  places  the  periods. 
Model  due  to  the  ambitious  energy  of  D.  F.  Graham. 


FIG.  238. 


FIG.  239. 

RETURN  OF  THE  FOLLOWER. 

In  most  cases  in  practical  machinery  it  is  very  essential  that  the 
return  of  the  follower  be  positive  in  order  to  avoid  breakages, 
though  sometimes  gravity  or  a  spring  may  be  employed  where  the 
velocity  is  not  unduly  high,  and  damage  not  likely  to  occur. 

That  the  Action  of  Gravity  may  Quicken  the  Return  of  the  end 
Dj  put  the  weight  TFnear  B,  Fig.  240,  and  make  the  arm  Z)JF  com- 
paratively light,  so  that  as  W  falls  by  gravity  the  end  D  will  move 
more  rapidly  than  W  by  reason  of  the  leverage. 

In  this  case  take#,  the  "  center  of  percussion  "  of  the  piece  BD. 


212  PRINCIPLES    OF    MECHANISM. 

When  g  is  in  a  horizontal  line  through  B,  g  starts  to  fall  just  as 


FIG.  240. 

fast  as  a  free  weight,  and  the  end  D  faster  in  proportion  to  its 
greater  distance  from  the  pivot  B. 

A  Positive  Return  is  readily  obtained  by  making  a  groove  for  the  end 


FIG.  241. 

D  to  run  in,  so  that  it  is  compelled  to  move  both  ways,  as  in  Fig.  241. 
A  positive  return  may  also  be  insured  by  means  of  an  extra  cam, 


FIG.  242. 

as  shown  in  Fig.  242,  one  cam  working  in  unison  with  the  other 
to  prevent  backlash  between  either  follower  and  its  cam. 


CAM   MOVEMENTS. 


213 


In  certain  cases  the  cam  may  be  made  of  constant  diameter 
with  the  follower  moving  in  a  straight  line,  as  shown  in  Fig.  243, 
when  the  follower  points  DD  may  just  contain  the  diameter  with- 
out backlash  whereby  the  follower  is  compelled  to  move  both  ways. 
This  may  serve  where  a  certain  law  of  motion  of  follower  is  to  be 
insisted  upon  for  only  a  half  revolution  of  A\ 
or  the  same  law  for  the  forward  and  return 
motion  of  follower  as  in  Figs.  222  to  226. 

In  cams  of  constant  breadth  the  return  of 
the  follower  may  be  assured,  as  in  Figs.  228, 
230,  236,  and  237. 

To  RELIEVE  FRICTION. 

Where  the  follower  is  compelled  to  drag 
over  the  full  extent  of  the  periphery  of  the  cam 
at  each  revolution  extensive  wear  is  very  likely 
to  occur.  Lubrication  will  help,  but  it  is  diffi- 
cult to  maintain  this  on  surfaces  so  exposed  to 
air  and  to  the  tendency  of  centrifugal  force 
to  throw  the  oil  off.  The  usual  remedy  is  to 
put  a  roller  at  D,  as  in  Fig.  244,  which  can 
track  along  on  the  exposed  edge  of  cam,  regardless  of  lubrication. 

While  rubbing  of  surfaces  occurs  at  the  pin  there  will  be  sur- 
face bearing,  instead  of  a  line  bearing  of  a  fixed  terminal  D  on  the 


B 


FIG.  244. 


edge  of  A.  Also  there  will  be  more  favorable  conditions  of  lubri- 
cation between  roller  and  pin  than  between  terminal  and  cam.  The 
amount  of  rubbing  action  of  surfaces  will  also  be  reduced  in  the 
ratio  of  the  diameter  of  roller  to  diameter  of  pin. 

In  conical  cams  the  roller  is  sometimes  made  conical. 


214 


PRINCIPLES   OF    MECHANISM. 


In  stamp  mills  a  disk  roller,  Fig.  245,  is  used,  which,  though 
largely  reducing  the  friction,  does  not  do  so  as  completely  as  the 
cylindric  roller.  Here  A  is  the  cam  lifting  the 
rod  EF  by  acting  against  the  collar  D  on  the 
rod.  The  rod  and  collar  both  turn  together  as 
A  may  dictate.  But  owing  to  the  thickness  of 
A,  with  D  rotating,  the  radius  of  D  at  one  side 
of  A  differs  from  that  of  the  other  side,  so  that 
there  is  a  combined  rolling  and  torsional  twist 
between  the  surfaces,  and  not  an  entire  relief 
from  slipping. 

A  flat-foot  follower,  as  in  Figs.  227  or  230, 
will  be  found  better  for  endurance  in  wear  than 
a  sharp  edge,  as  in  Fig.  243. 


MODIFICATION  OF  CAM  REQUIRED  BY 
FOLLOWER. 

The  immediate  result  of  solution  of  a  cam 
motion  is  usually  a  so-called  linear  cam,  from 
which  the  practical  cam  is  to  be  obtained. 
In  Fig.  246  the  linear  or  theoretical  cam  is  the  curve  Dabc, 
etc.,  which  is  the  line  as  traced  on  A,  which  the  theoretical  fol- 
lower point  D  is  to  follow  as  A  revolves. 

To  Introduce  a  Follower  Roller  at  D  and  maintain  the  theoretical 
action,  the  center  point  of  the  roller  must  follow  the  linear  cam. 


E 


FIG.  245. 


FIG.  246. 

Dabc.  To  insure  this,  strike  circles  equalling  the  roller  in  diameter 
at  numerous  points  along  the  linear  cam,  as  at  «,  b,  c,  etc.  Then 
draw  the  practical  cam  lines  tangent  to  these  circles  throughout. 


CAM    MOVEMENTS. 


215 


If  these  enveloping  cam  curves  are  drawn  at  both  sides  of  these 
circles,  we  have  a  drawing  of  the  cam  groove  for  cam  A,  providing 
for  a  positive  return  of  the  cam  follower. 

In  practice,  this  groove  is  best  executed  in  metal,  by  using  a 
cutter  of  the  same  shape  and  diameter  as  the  roller  D,  and  so 
mounting  it  in  a  cutting  machine  that  the  cutter  while  cutting  is 
compelled  to  move  with  its  center  following  the  line  abc,  etc. 

In  Cams  with  Salient  Angles,  as  in  Fig.  219  and  the  heart  cam  of 
Fig.  222,  a  sacrifice  must  be  made  from  the  theoretical  action  of 
the  cam  by  the  introduction  of 
a  roller,  as  shown  in  Fig.  247. 
Let  FNEMG  represent  the  lin- 
ear or  theoretical  cam  and  the 
line  the  center  of  the  roller 
should  follow.  Drawing  a  series 
of  roller  circles,  and  the  practi- 
cal cam  HIJ  tangent  to  the  cir- 
cles, it  is  found  that  at  /  the 
cam  is  foreshortened  by  the 
amount  LE.  That  is,  for  the 
practical  cam  HI  to  compel  the 
roller  center  to  move  on  the 
linear  cam  FN  to  E,  the  cam  FlG  247 

surface  HI  would  require  to  be 

extended  to  some  point  near  M,  where  its  normal  would  strike  E. 
But  as  the  cam  is  cut  away  beyond  /  by  U,  the  roller  is  at  liberty 

to  swing  around  its  point  of  con- 
tact with  /,  the  center  of  the 
roller  describing  the  circle  NLM 
and  failing  to  reach  the  point  E 
in  the  linear  cam  by  the  amount 
LE. 

This    circumstance    may    at 
times    prohibit   the   use  of   the 
antifriction  roller,  as  depending 
upon  how  essential  the  shortage 
EL  may  be.       Or,  at  times,  the 
roller  may  be  used  of  diameter 
smaller  than  otherwise  preferred. 
For  a  Thickened  and  Rounded  Follower  Extremity,  D,  as  in  Fig. 
248,  where  some  point  D  is  to  follow  the  linear  cam  as  shown,  cut 


-  248- 


216 


PRINCIPLES    OF    MECHANISM. 


a  templet  BD  of  the  right  shape  at  D  for  the  follower,  and  with 
the  point  D  notched  by  a  V  notch  as  shown,  through  which 
the  linear  cam  can  be  seen.  Then,  with  a  circle  about  A 
through  B,  place  the  templet  at  numerous  positions,  as  at  Br D' , 
and  draw  a  curve  at  the  end,  as  shown.  A  cam  line,  traced  tangent 
to  all  these  curves  as  shown,  will  be  the  practical  cam  required. 

For  an  Edge  Cam,  as  in  Figs.  242  and  244,  and  for  a  disk  or  face 
cam,  with  a  cam  groove  in  the  side,  as  in  Fig.  246,  the  roller  should 
be  cylmdric  without  question. 

The  Best  Form  of  Roller  has  been  a  matter  of  some  question,  as 
some  designers  use  a  cylindric  and  others  a  conic  one  for  a  cylm- 
dric cam,  such  as  shown  in  Figs.  214  and  215. 

With  regard  to  the  action  on  the  outside  surface  of  the  roller 
alone,  it  appears  that  when  the  roller  is  moving  longitudinally  in 


A 


b\ 


Fm.  249, 

its  groove,  if  such  might  be,  its  form  should  be  the  cylinder;  while 
when  the  cam  is  revolving  and  the  roller  nearly  stationary  its  form 
should  be  conic,  with  the  vertex  of  the  cone  at  the  axis  of  the  cam. 
For  forms  of  the  cam  between  the  above  limits  the  roller  would 
seem  to  require  some  compromise  form  as  between  the  above  cylin- 
der and  cone.  A  little  consideration  will  show,  however,  that  a 
perfect  form  for  simple  rolling  contact  in  this  case,  of  axes  of  cam 
and  roller  meeting,  does  not  exist. 

Suppose  the  cam  approaches  one  of  the  same  velocity-ratio  for- 
ward and  back,  as  in  Fig.  249,  where  A  A'  is  the  cylmdric  cam  and 
EF  its  development.  Take  ab  for  one  side  of  the  cam  groove, 
and  de  the  same  in  the  development. 

Now,  if  the  roller  is  made  conical  with  its  vertex  at  the  axis  A, 
when  the  roller  rolls  from  a  to  b  at  the  surface  of  the  cam  it 


CAM    MOVEMENTS.  217 

/ 

f 

would  be  necessary  for  the  vertex  to  roll  from  a  to  c  on  the  axis  A, 
which  is  clearly  impossible  since  the  vertex  is  a  point.  From  this 
it  would  seem  that  the  roller  must  have  some  size  at  the  axis;  and 
probably  the  best  that  can  be  done  by  approximating  it  is  to  lay  off 
the  distance  ac  atfg,  where  fd  equals  the  radius  of  the  cylinder  A; 
draw  df  and  eg  produced  to  meet  in  0,  and  take  dO  as  the  length 
of  the  cone  from  the  large  end  of  which  to  cut  the  follower  roller. 

Perfect  rolling  of  the  follower  roller  rolling  along  a  cam  surface 
ab  would  require  that  the  axes  of  cam  and  roller  pass  each  other 
with  a  distance  between,  and  that  the  roller  be  a  hyperboloid  of 
revolution;  also,  that  the  distance  between  the  axes  varies  as  the  in- 
clination ab  of  the  cam  curve  varies.  It  therefore  seems  impossible 
to  obtain  a  perfect  follower  roller  for  a  cylindric  cam,  that  is,  one 
where  the  outside  surface  of  the  roller  simply  rolls  on  the  surface 
of  the  cam  groove,  because  the  action  will  of  necessity  be  partly 
rolling  and  partly  torsional  slip  of  surfaces,  and  vary  with  slope  of  ab. 

This  torsional  slip  of  surfaces  will  be  the  same  for  a  truncated 
conic  roller  with  vertex  at  axis  of  cam  and  moving  in  a  cam  groove 
parallel  to  the  axis,  as  for  a  cylindric  roller  of  equal  length  moving 
in  a  groove  encircling  the  cam;  and  according  to  Fig.  249  these 
torsional  slips  tor  both  rollers  will  be  alike  for  a  groove  at  an  angle 
of  30  degrees  with  the  circles  of  the  cam  surface. 

The  Action  of  the  Holler  upon  the  Pin  and  its  Shoulders  is  of 
importance,  as  it  is  easily  seen  that  the  conic  roller  will  be  severely 
pressed  against  its  shoulder,  causing  friction  on  a  surface  of  larger 
diameter  than  the  pin,  thus  introducing  a  very  serious  resistance  to 
rotation  of  roller — an  objection  which,  probably,  by  far  outweighs 
the  advantage  of  a  conic  roller,  except  in  cases  where  the  cam  is  of 
large  diameter  relative  to  the  throw. 

With  regard  to  the  pin  for  supporting  the  roller:  when  it  can 
be  made  conical,  with  the  same  angle  of  convergence  as  the  conic 
roller  itself,  the  end  thrust  causing  shoulder  friction  will  be  mostly 
avoided,  and  the  conic  roller  will  be  much  more  acceptable.  One 
drawback  here,  however,  may  be  found  in  the  large  average  size  of 
pin,  and  it  may  be  advisable  to  make  it  part  way  cylindric  and  the 
remainder  conic. 

For  a  Conic  Cam  the  question  of  best  form  of  follower  roller  is 
still  more  involved,  and  it  is  probably  advisable  to  adopt  the  cylin- 
dric one. 

For  the  Spherical  Cam  there  seems  to  be  no  question  but  that 
the  roller  should  be  conical,  since  here  the  vertex  of  the  cone  can 


218  PRINCIPLES    OF    MECHANISM. 

remain  at  the  exact  center  of  the  sphere,  and  there  will  be  theoreti- 
cally perfect  rolling  between  the  roller  and  its  cam  groove.  But 
even  here  the  shoulder  friction  due  to  the  endlong  thrust  will  be 
found  a  serious  objection  to  the  conic  form  of  roller  unless  the 
axial  pin  for  the  roller  can  also  be  conical  to  match,  or  partly  conic 
and  partly  cylindric. 


CHAPTER  XIX. 
INVERSE  CAMS  AND  COUPLINGS. 

I.    THE  INVERSE  CAM. 

THIS  term  may  be  given  to  a  movement  which  has  the  elements 
of  a  grooved  cam  and  follower,  but  where  the  driver  has  the  pin  or 
roller  and  where  the  follower  has  the  groove;  styled  by  Willis  the 
pin  and  slit. 

The  inversion  of  the  movement  is  to 
avoid  dead  points  that  would  in  some 
cases  occur  when  used  as  a  cam  move- 
ment. 

The  peculiarity  which  distinguishes 
this  from  the  cam  movement  consists 
in  the  fact  that  here  the  pin  recipro- 
cates in  the  slot  or  groove,  while  in  the 
cam  it  does  not. 

The  slotted  piece  B,  Fig.  250,  is  the  FIG.  250. 

follower  and  A  the  driver  with  directional  relation  constant.      The 
velocity-ratio  as  in  cams  is 

ang.  velocity  of  A  _BG 
ang.  velocity  of  B~~  AC' 

The  slot  may  be  made  straight  on  a  radial  line  or  not,  and  a 
block  may  be  fitted  on  the  pin  and  in  the  slot  to  avoid  wear  by 
extending  the  bearing  surface.  Thus  equipped  with  the  slot  radial, 
this  movement  is  sometimes  known  as  the  Whiteworth  Quick  Re- 
turn. It  is  considerably  used  on  English  shaping  machines. 

An  example  of  this  movement  for  directional  relation  changing 
is  given  in  Fig.  251.  In  this  there  is  a  point  of  tarrying  of  the 
driven  piece  by  reason  of  the  slot  having  quite  a  portion  made  to 
the  circular  path  of  the  driver.  Thus  by  curving  the  slot,  modi- 
fications of  motion  may  be  obtained. 

This  movement  has  been  used  to  give  motion  to  the  needle-bar  of 
a  sewing-machine. 

219 


220 


PRINCIPLES    OF    MECHANISM. 


Another  example  is  given  in  Fig.  252  of  a  needle-bar  cam 
motion  much  used  in  sewing-machines.  It  corresponds  somewhat 
with  Fig.  251,  except  that  the  axis  B  is 
in  effect  removed  to  infinity  by  the 
mounting  of  EF  on  a  sliding  bar. 

At  F  the  slot  may  be  made  to  cor- 
respond with  the  circle  arc  described 
by  the  pin;  so  that  the  needle  will 


FIG.  251. 


FIG.  252. 


stand  stationary  while  the  shuttle  passes  the  loop  of  thread.     To 
form  the  initial  loop,  a  quirk  may  be  introduced  in  the  slot  at  E. 

In  some  applications  in  heavy  machinery,  this  slotted  piece  EF 
has  a  straight  slot  and  a  block  on  the  pin  fitted  to  slide  in  the  slot. 
Thus  the  pitman  has  been  avoided  in  steam-engines  and  the  engine 
correspondingly  shortened. 

By  a  sufficiently  wide  slot  in  FE,  the  movement  may  be  placed 

at  an  intermediate  point  on  a  straight 
shaft  and  eccentric. 

II.  COUPLINGS  BY  SLIDING  CONTACT. 

Oldham's  Movement  with  direc- 
tional relation  and  velocity-ratio 
constant  is  illustrated  in  elementary 
form  in  Fig.  253  for  connecting 
axes  that  are  parallel  but  not  co- 
incident, acting  by  sliding  contact. 

The  velocity-ratio  is  constantly 
equal  to  1,  as  easily  seen  in  the  small 


diagram  of  a  section  normal  to  the  shafts.     The  arms  of  the  con- 
necting cross  which  slide  in  the  sockets  made  fast  upon  the  shafts 


INVERSE    CAMS    AND    COUPLINGS. 


221 


must  constantly  pass  through  both  of  the  axial  points  A  and  B-, 
and  as  they  form  a  right  angle,  the  point  of  intersection  of  the 
cross  must  follow  the  circle  of  diameter  AB  as  shown,  since  all 
lines  at  right  angles  drawn  from  the  extremities  of  a  diameter 
meet  in  the  circle  to  that  diameter.  The  extent  of  sliding  per 
revolution  on  each  branch  of  the  cross  equals  two  diameters,  AB. 

If  the  cross  is  not  right-angled,  the  same  is  true  of  the  angular 
velocity,  as  shown  in  Fig.  254 "  but  the 
amount  of  sliding   is   greater,  since  the 
circle  is  thrown  to  one  side  and  increased 
in  diameter. 

A  Complication  of  Movement  results 
from  placing  the  axes  out  of  parallel,  as  in 
Fig  255,  so  that  they  meet  at  some  point,  A 
0.  Then  ED  shows  one  position  of  the 
cross,  the  angle  being  at  E  and  F.  An- 
other position,  a  quarter-turn  away,  is 


FIG.  254. 


FIG.  255. 


shown  at  HI,  with  the  angle  at  /  and  /.  These  diagrams,  com- 
pared, show  that  the  shaft  B  has  endlong  motion  to  the  extent 
Gff,  twice  in  a  revolution,  regarding  A  as  without  end  play.  The 
angle  point  of  the  cross,  however,  always  remains  on  the  line  1G, 
or  plane  normal  to  A,  and  describes  a  circle  KLM  in  that  plane. 

To  study  the  relative  motions  of  the  shafts  A  and  B,  take  the 
cross  as  right-angled  and  in  an  intermediate  position,  KL,  LM. 
The  ^l-branch  will  always  be  found  in  the  plane  PN,  normal  to  A, 
and  the  .S-branch  in  a  plane  NQ,  normal  to  B.  These  planes  will 
intersect  in  a  line  perpendicular  to  the  plane  of  the  axes  A  and  B, 
or  in  some  line  N,  RL,  for  the  position  of  the  cross  as  shown. 
Draw  a  circle  about  L  on  the  plane  normal  to  A,  and  an  ellipse  to 
the  same  center,  to  represent  a  circle  on  the  plane  normal  to  B. 
Taking  KL  for  the  A  -branch  of  the  cross,  and,  as  in  the  plane  of 
the  paper,  the  ^-branch  in  its  own  plane  will  appear  at  ML  with- 


222 


PRINCIPLES   OF    MECHANISM. 


MLK  a  right  angle  because  lines  at  right  angles  in  space  will 
appear  at  right  angles  in  projection,  when  one  of  the  lines  is  paral- 
lel to  the  paper.  Hence,  if  we  move  B  from  parallel  to  A  into  its 


FIG.  256. 


inclined  position,  while  A  and  KL  remain  fixed,  the  point  b  in  the 
circle  must  move  to  c  in  the  ellipse.  To  determine  the  angular  dis- 
turbance of  B  in  this  movement,  swing  B  and  its  point,  c  in  the 
ellipse,  back,  without  angular  disturbance,  to  parallel  with  .4,  when 
the  ellipse  returns  to  the  circle  and  the  point  c  will  fall  at  d,  cd 
being  perpendicular  to  RL. 

Then  cLd  will  be  the  change  in  the  angular  position  of  the 
.Z?-arm  of  the  cross  as  due  to  the  swinging  of  B  from  parallel  to  A, 
to  the  position  BO. 

Let  x  and  y  represent  the  angles  SLd  and  SLb  respectively 
and  a  the  angle  between  the  axes  A  and  B.  Then  Fig.  256  will 
show  the  relation  of  these  angles,  from  which  we  get  EN  = 
DN  cos  a,  EN  =  LS  tan  y,  DN  =  LS  tan  x,  which  gives  us  the 
relation 

tan  it 

cos  a  =  -  — -  , 
tan  x 

in  which,  a  being  constant,  y  may  be  found  when  x  is  given,  or  x 
found  when  y  for  any  point  in  the  revolution  is  given. 

It  may  be  noted  that  the  same  figure  in  cross-section  at  RKLM 
is  obtained  whether  the  intersection  is  at  0  or  at  P;  also  the  same 
equation.  But  it  is  clear  that  when  0  is  at  P  the  movement  re- 
duces to  that  of  the  Hooke's  universal  joint,  which  is  free  from  the 
sliding  motion  of  Fig.  255,  and  hence  the  latter  can  claim  no  ad- 
vantages over  the  Hooke's  joint. 

The  velocity-ratio   could  be   found  by  calculating  a  series   of 


INVERSE   CAMS   AND   COUPLINGS.  223 

angular  positions  from  the  formula  and   comparing  them.     But 
this  is  best  done  by  differentiating  the  formula  and  obtaining 

ang.  veloc.  A       dx      cos2  x 
velocity-ratio  =  -  — ~  =  -7-  =  — —,  sec  a 

ang.  veloc.  B      ay      cos  y 

_  1  —  sin2  x  sin2  a 
cos  a 
cos  a 


"1  —  cos'2  y  sin2  a 

When  a  equals  0  the  angular  velocities  are  equal,  as  they  evi- 
dently should  be,  and  the  velocity-ratio  =  1. 

These  equations  are  the  same  as  given  by  Willis,  p.  452,  for 
Hooke's  joint. 

For  a  =  45  degrees;   -,-  =   4/2,  or  =  -~t   for  maximum  and 
ay  & 

minimum  values  as  occurring  for  x  —  0  and  90  degrees  respectively. 

dx 
The  velocity-ratio  is  j— =  1  for  tan2  y  =  cos  a  =  cot2  x.     These 

equations  are  the  same  as  found  by  Willis  and  Poncelet  for  the 
Hooke's  joint. 

A  Peculiar  Movement  transmitting  motion  from  A  to  B  is  illus- 
trated in  Fig.  257,  in  which  the  contact  between  the  parts  is  by 
sliding,  except  where  the  axes  are  in  direct  line.  In  the  model, 
the  axes  may  be  arranged  parallel  or  at  various  angles  and  at  var- 


FIG.  257. 


ions  offsets,  as  in  Figs.  253  and  255,  and  it  appears  to  be  one  way 
of  realizing  those  cases  in  material  form,  except  that  here  the  axis 
P>  is  not  compelled  to  slide  endwise. 

To  study  the  velocity-ratio  it  is  most  convenient  to  imagine  the 
intermediate  slotted  piece  to  be  replaced  by  a  cross  the  branches 


224 


PRINCIPLES   OF   MECHANISM. 


of  which  are  perpendicular  to  the  slots  they  stand  for.  The  move- 
ment will  then  serve  for  any  and  all  the  conditions  brought  out  in 
Fig.  255  or  253  and  with  the  same  law  of  velocity-ratio. 

These  joints  all  have  two  points  of  maximum  and  two  points  of 
minimum  velocity  in  each  revolution,  as  also  a  pair  of  the  two-lobed 
elliptic  wheels  of  Fig.  44,  but  the  law  of  velocity-ratio  of  the  latter 
is  different. 


FIG.  258. 

When  the  movement  of  Fig.  257  is  set  with  axes  parallel,  it  acts 
the  same  as  the  three  disks  in  Fig.  258,  which  latter  movement 
was  used  by  Oldham  in  appliances  employed  in  the  Banks  of  Eng- 
land, by  Winan  in  his  Cigar  Boat,  and  also  by  C.  T.  Porter  to  couple 
the  shafts  of  an  engine  and  dynamo  nominally  in  line,  but  practic- 
ally a  little  "  off  line  "  by  reason  of  temperature,  flexure,  etc. 


CHAPTER  XX. 
ESCAPEMENTS. 

DIRECTIONAL   RELATION    CHANGING.      VELOCITY-RATIO 
CONSTANT   OR   VARYING. 

AN  escapement  is  a  movement  in  which  the  follower  is  driven  a 
distance,  usually  by  sliding  contact,  to  where  the  driver  is  allowed 
to  pass  free  for  a  little  space,  when  another  engagement  by  sliding 
contact  occurs  to  drive  the  follower  back  to  a  position  for  repeating 
an  engagement  like  the  first. 

POWER  ESCAPEMENTS. 

Fig.  259  illustrates  an  escapement  of  a  design  suitable  for  use  in 
heavy  machinery  in  which  the  motion  is  continuous  and  where  there 


FIG.  259. 

is  no  lock  such  as  used  in  most  cases  of  escapements  for  clocks  and 
watches. 

As  F  escapes  from  E,  the  arm  G  should  be  near  to  D  to  prompt- 
ly engage;  and  the  point  of  initial  contact  at  D  should  be  as  near 
to  the  longitudinal  line  through  A  as  possible,  to  relieve  the  blow 
due  to  initial  contact. 

Escapements  are  mostly  employed  in  timepieces,  and  should  be 
as  nearly  as  possible  such  as  to  give  to  the  vibrating  pendulum  or 
balance  equal  impulses  at  all  times,  and  be  as  free  as  possible  from 
hindering  the  vibration  by  frictional  contact  of  parts  with  the  pen- 
dulum or  balance.  That  is,  the  higher  essentials  for  fine  time- 
keeping are,  1st,  an  isodynamic  or  equal-impulse  escapement;  2d, 
an  isochronous  pendulum  or  balance;  and  3d,  freedom  of  vibrating 

225 


226 


PRINCIPLES   OF   MECHANISM. 


parts  from  contact  with  the  other  pieces,  except  when  receiving  the 
impulses.  There  are  other  considerations,  such  as  temperature, 
position,  etc.,  which  are  outside  of  our  present  topic. 

THE  ANCHOR  ESCAPEMENT. 

In  Fig.  260  is  shown  a  so-called  anchor  escapement,  the  name 
being  due  to  the  resemblance  of  the  vibrating  piece  to  the  ship's 
anchor. 

As  shown  in  full  lines,  we  have  the  dead-beat  escapement,  in 
which  the  escape-wheel  A  stands  still  while  a  tooth  rests  on  a  pallet, 
notwithstanding  the  movement  of  the  pallet.  As  shown,  the  tooth 
1  is  about  to  move  forward  upon  the  pallet  HG9  as  the  latter  ad- 


FIG.  260. 

vances  left-handed.  Reaching  ff,  a  further  movement  of  H  to 
the  right  and  return  allows  7  to  remain  stationary,  because  the 
pallet  from  H  back  is  formed  to  a  circle  arc  about  the  center  B. 
A  like  action  occurs  when  the  tooth  rests  and  slides  upon  the  pallet 
E.  The  teeth  of  A  should  be  so  formed  and  cut  away  as  to  permit 
the  pallet  to  move  a  considerable  distance  after  the  escapement  of 
a  tooth  and  before  the  return  of  the  pallet  occurs,  so  that  the  pen- 
dulum may  complete  its  swing. 


ESCAPEMENTS.  227 

Thus  the  tooth  is  just  on  the  point  of  escaping  at  D,  following 
which  /  moves  forward  upon  the  pallet  .//as  the  latter  swings  to  the 
right,  allowance  being  made  for  HG  to  move  still  farther  toward  A. 
As  HG  returns,  the  pressure  of  the  tooth  /  upon  the  slope  HG  im- 
parts an  impulse  to  GHB  toward  the  left.  Similarly  at  />,  the  tooth 
is  just  completing  its  impulse  on  ED  toward  the  right.  These  im- 
pulses overcome  the  retarding  influences  of  the  air  and  other  re- 
sistances acting  upon  the  pendulum,  thus  maintaining  its  motion. 

To  construct  this  movement,  the  pallets  are  somewhat  thinner 
than  the  half  of  the  pitch  of  the  escape-wheel,  so  as  to  give  a  slight 
drop  ///  to  insure  the  landing  of  /  upon  H  at  a  slight  distance  from 
the  bevel  HG.  From  H  and  G  back,  the  pallets  are  formed  of  cir- 
cle arcs  struck  from  the  center  B.  The  bevel  HG  may  be  as- 
sumed by  a  line  drawn  to  J  tangent  to  a  circle  struck  from  B.  The 
pallet  ED  should  be  formed  to  the  same  circles  as  HG,  and  beveled 
by  a  tangent  to  the  same  circle  JK.  Then  the  angle  DEE  will 
equal  the  angle  HBG,  as  it  evidently  should  to  balance  the  im- 
pulses. 

It  has  been  proposed  to  put  the  impulse  bevel  upon  the  teeth  of 
the  escape-wheel  instead  of  the  pallets.  Fig.  260  would  nearly  fit 
the  case  by  turning  A  backwards  when,  as  the  tooth  of  D  escapes, 
the  tooth  MN  would  fall  upon  the  circle  of  the  pallet  G  as  an  arc 
of  repose,  until,  when  the  pallet  returns,  it  would  receive  an  im- 
pulse in  sliding  along  the  bevel  NM  until,  when  M  escapes  at  G,  0 
of  the  next  tooth  OP  will  laud  upon  the  pallet  />,  and  in  due  time 
impart  an  impulse  from  the  bevel  OP. 

Again,  the  impulse  bevel  may  be  divided  between  the  pallet  and 
the  teeth  of  the  escape-wheel,  as  really  done  in  the  ordinary  "  lever 
escapement "  of  watches. 

TJie  recoil  escapement  is  obtained  from  the  above  by  producing 
the  bevel  lines  HG  and  DE,  as  shown  in  dotted  lines,  and  modify- 
ing the  teeth  of  the  wheel  to  some  shape  as  dotted  at  /.  Then,  as 
the  tooth  at  D  escapes  from  DEF,  the  tooth  I  will  strike  upon  the 
bevel  GHL,  and,  as  the  pendulum  moves  still  farther  in  the  same 
direction  before  returning,  /will  be  forced  'to  slide  up  towards  L, 
thus  giving  to  the  escape-wheel  a  slight  backward  movement,  called 
the  recoil. 

The  recoil  escapement  is  the  most  common  one  in  ordinary  man- 
tel clocks,  and  regarded  as  inferior  to  the  dead-beat,  which  latter  is 
usually  introduced  into  the  finer  mantel  clocks,  regulators,  astro- 
nomical and  many  tower  clocks. 


228 


PRINCIPLES  OF  MECHANISM. 
THE  PIN-WHEEL  ESCAPEMENT. 


This  is  so  named  because  the  escape-wheel  has  pin  teeth,  and  it 
is  shown  in  Fig.  261. 

The  pallets  are  formed  to  circle  arcs  ej,fi,  gl,  and  lik,  terminated 
with  bevels  ef  and  gli  upon  which  the  pins  slide  to  impart  the  im- 
pulses to  the  pallets.  A  pin  of  the  escape-wheel  is  shown  as  resting 
upon  the  pallet  lik.  If  the  pallets  are  moving  toward  the  left,  the 
pin  slides  upon  the  circle  arc,  or  "  arc  of  repose,"  toward  k  until  the 


FIG.  261. 

pendulum  reaches  its  limit  of  swing,  when  it  returns,  and  also  the- 
pallet,  which,  on  arriving  at  the  pin,  permits  the  latter  to  slide 
down  the  bevel  hg,  imparting  the  impulse.  It  escapes  from  the 
pallet  glik  and  alights  upon  fi,  of  the  pallet  efi.  The  pendulum 
completes  its  movement  and  returns,  when  the  pin  slides  down 
fe  and  imparts  a  second  impulse  opposite  to  the  first.  As  the  pin 
escapes  at  e,  the  next  pin  drops  upon  hk,  to  repeat  the  movements 
of  the  former  one,  etc. 

To  construct  the  movement,  draw  circles  from  the  center  B 
for  the  pallets,  the  latter  having  thicknesses  such  that  the  two, 
with  a  pin  between,  will  swing  between  two  adjacent  pins  of  the 
escape-wheel,  with  a  necessary  slight  clearance.  Then  the  bevels 
ef  and  gU  are  so  determined  as  to  subtend  the  same  arbitrary  angle 
at  B  as  bBc,  besides  allowing  a  small  safety  angle  ab  equal  to  cd,  as 
providing  for  the  distance  from  the  impulse  bevel  back  to  the  land- 
ing point  of  the  pin  upon  the  pallets.  This  angle  may  be  small 
and  possibly  zero  for  cylindrical  pins.  When  the  pin  escapes  at  e 


ESCAPEMENTS.  229 

(see  dotted  line),  the  next  pin  should  land  upon  lik  at  the  allowed 
angle  aBb  from  h.  Also,  when  the  pin  escapes  from  hg  (see  dotted 
line  struck  from  D  with  a  radius  equal  to  distance  from  A  to  inside 
of  pin)  it  should  drop  upon//  at  the  same  angle  cBd  equal  to  a£b, 
from  the  initial  point /of  the  bevel. 

This  escapement  has  been  considerably  employed  for  tower 
clocks,  and  has  the  advantage  that  the  pins  may  conveniently  be 
hardened,  or  even  made  of  glass  rod  or  cut  jewel  stones.  In  some 
cases  the  upper  half  of  the  pins  are  cut  away,  since  this  portion  is 
not  acted  upon  by  the  pallets,  thus  permitting  the  placing  of  adja- 
cent pins  a  whole  pin  diameter  closer.  Then  with  the  same  num- 
ber of  pins  and  thickness  of  pallets,  the  wheel  A  will  be  made 
smaller,  and  the  strength  of  an  impulse  will  be  materially  increased 
other  things  the  same.  Also,  the  drop  of  a  pin  on  escaping  will  be 
reduced  by  nearly  a  half,  and  the  "  tick "  will  be  materially 
quieted. 

THE  GRAVITY  ESCAPEMENT. 

This  name  is  given  to  escapements  where  the  fall  of  a  weight 
through  a  definite  height  imparts  the  impulse,  all  impulses  being 
thus  equalized  in  intensity.  A  spring  may  be  used  instead  of 
gravity  to  measure  the  definite  impulse.  Such  are  sometimes  called 
isodynamic  escapements. 

A  gravity  escapement  employed  by  Wm.  Bond  &  Sons  in 
chronographs,  and  called  isodynamic,  is  illustrated  in  Fig.  262. 
The  same  has  been  used  in  tower  clocks  with  good  results,  and  it  is 
believed  to  be  about  the  best  in  kind  and  in  construction  for  that 
purpose,  being  the  same  in  principle  as  the  Bloxam's  or  Dennison's. 

At  A  is  the  driving  axis,  on  which  is  made  fast  a  collar  with  an 
eccentric  pin,  a,  and  an  arm,  GH.  A  T-shaped  gravity  piece,  DEF, 
is  suspended  by  a  spring,  N,  from  a  fixed  clamp,  QR,  and  will  swing 
right  and  left,  and  it  may  rest  against  the  eccentric  pin  a,  or  it  may 
rest  by  its  pin  at  F  against  the  pendulum  rod  P.  At  D  is  a  pin 
flattened  on  the  lower  side  normal  to  a  line  DN.  A  second  T- 
shaped  gravity  piece,  IMK,  is  suspended  on  the  other  side  and 
like  the  first  except  they  are  rights  and  lefts,  and  the  pin  at  I  is 
flattened  on  the  upper  side  normal  to  a  line  IS. 

The  pendulum  rod  P  is  suspended  from  the  clamp  QR  by  a 
spring,  not  shown,  which  allows  it  to  swing  to  the  right  and  left. 

The  eccentric  pin  a  throws  the  gravity  pieces  to  the  right  or 
left.  As  shown,  this  pin  a  holds  the  gravity  piece  DEF  in  the 


230 


PRINCIPLES   OF   MECHANISM. 


extreme  position  the  pin  can  give  it ;  and  the  shaft  A  is  locked  in 
that  position  by  the  end  of  the  slender  arm  GH  striking  against  the 
detent  pin  at  D.  Now,  as  the  pendulum  rod,  moving  toward  the 
left,  strikes  the  pin  at  F,  the  gravity  piece  DEF  is  carried  along 
with  the  pendulum  to  its  limit  of  movement,  thus  releasing  the 
arm  GH,  when,  on  making  a  half -turn,  it  is  arrested  by  the  end  of 
the  arm  GH  meeting  the  detent  pin  /,  which  now  will  be  in  the 


Q 


N 


D 


R 


M 


K 


FIG.  262. 

dotted  position  <7,  because  the  eccentric  pin  a  has  moved  with 
H  to  the  dotted  position  b  and  thrown  the  gravity  piece  I  KM  into 
the  position  JLM.  This  detent  pin  /  detains  the  arm  GH  till  the 
pendulum  rod,  on  returning  from  its  extreme  position  to  the  left, 
meets  the  pin  L,  and  carries  JLM  along  with  it,  thus  releasing  the 
arm  Hfaom  the  detent  pin  J,  when  the  arm,  the  shaft  A,  and  the 
eccentric  pin  a  make  another  half-turn. 


ESCAPEMENTS.  231 

Now  it  is  readily  seen  that  as  the  pendulum  rod  returns,  it  is 
followed  by  JLM  to  the  position  IKM.  That  is,  the  pendulum  rod 
takes  the  gravity  piece  from  JLM  to  the  limit  of  movement  and 
back  to  IKM,  one  operation  neutralizing  the  other  as  far  back  as  to 
JLM\  but  from  the  excess  movement  JLM  to  IKM  the  pendulum 
receives  its  impulse.  A  like  impulse  is  received  from  F  in  the 
opposite  movement  of  the  pendulum.  The  eccentric  pin  «,  moved 
by  the  train  of  wheelwork,  raises  the  gravity  piece  from  K  to  L9 
or  lifts  its  center  of  gravity  to  create  the  impulse. 

The  impulses  imparted  to  the  pendulum  are  thus  made  practi- 
cally equal  for  all  time  of  running  of  the  clock. 

THE  CYLINDER  ESCAPEMENT. 

The  cylinder  escapement,  formerly  much  used  in  Swiss  watches, 
and  interesting  from  the  standpoint  of  mechanism,  is  illustrated  in 


FIG.  263. 

Fig.  263.  At  A  is  the  driving  staff  carrying  the  escape-wheel  with 
peculiar  shaped  teeth  having  inclined  edges,  CD.  These  teeth  act 
upon  pallets  at  C  and  E,  consisting  of  the  smoothed  edges  of  a 
thin  half -cylinder,  CPE,  supported  to  swing  upon  a  central  axis,  B. 
The  diameter  of  the  inside  of  this  cylinder  is  just  sufficient  to  allow  » 
the  latter  to  swing  over  the  tooth  CD  with  a  trifle  of  clearance, 
and  the  distance  CH,  equal  to  1G,  should  be  just  sufficient  to  admit 
the  full  cylinder  with  a  trifle  of  clearance. 

The  tooth  CD  is  represented  as  escaping  from  the  edge  of  the 
cylinder  at  C  and  soon  to  strike  just  inside  the  edge  at  E  and  to 
slide  some  distance,  the  cylinder  turning  right-handed.  On 
reaching  the  limit  of  movement  E  returns,  and  on  arriving  at  D 


232 


PRINCIPLES   OF   MECHANISM. 


the  inclined  edge  DC  of  the  tooth  will  slide  against  the  edge  E, 
imparting  a  left-handed  impulse  to  the  cylinder  B.  As  the  tooth 
CD  escapes  from  E,  the  point  H  will  strike  just  back  of  the  edge 
of  the  cylinder  at  G,  and  slide  on  the  outside  to  the  limit  of  move- 
ment, when  on  returning,  as  the  edge  G  passes  H,  the  inclined 
tooth  HI  will  slide  along  against  the  edge  G  and  impart  a  right- 
handed  impulse.  On  completing  this, 
CD  escapes  at  C  as  before.  The 
amount  of  inclination  of  CD  is  arbi- 
trary, and  it  may  be  straight  or  some- 
what convex. 

There  seems  to  be  a  large  amount 
of  friction  in  this  escapement,  parts 
being  practically  the  whole  time  in 
rubbing  contact,  which  circumstance 
may  be  sufficient  to  explain  the  un- 
spirited  deportment  of  the  escape- 
ment as  observed  in  the  watch. 
THE  LEVER  ESCAPEMENT. 
This  escapement,  Fig.  264,  is  the 
one  most  used  in  watches  and  known 
also  as  the  "  anchor  escapement," 
"  detached  escapement,"  or  "  detached 
lever  escapement,"  because  BH  is 
a  lever,  or  because  DEBH  resem- 
bles the  anchor,  or  because  the  balance  parts  JKO  are  detached 
and  free,  respectively. 

This  leaves  the  vibrating  balance  freest  from  friction  of  all  the 
escapements,  unless  it  be  that  for  chronometers.  As  compared  with 
Fig.  263,  it  has  a  very  decided  advantage  in  this  respect,  a  fact 
evinced  by  its  more  lively  action  as  observed  in  watches. 

In  this  escapement  A  is  the  axis  of  the  escape-wheel,  D  and  E 
pallets  of  the  anchor-piece  DEBH  swinging  about  an  axis  at  B. 
The  balance  staff  is  at  0,  upon  which  is  a  collar  holding  a  pin  at  J, 
at  which  the  collar  is  cut  away  nearly  to  the  pin.  The  lever  is 
slotted  at  /  and  has  a  pin  or  shoulder  at  H.  At  L  and  M  are  bank- 
ing pins  or  other  provision  against  over-movement  of  the  lever.  In 
this  there  are  really  two  movements,  one  the  escapement  proper 
between  A  and  B,  and  the  other  a  pin  and  slit  motion  between  B 
and  0.  Also  a  lock  as  between  the  pin  H  and  the  edge  of  the 
collar,  to  prevent  BH  from  moving  except  at  the  proper  time. 


FIG.  264. 


ESCAPEMENTS.  233 

Now  supposing  J  to  be  moving  toward  /  by  the  swing  of  the 
balance,  when  the  pin  J  enters  the  slit,  the  pin  H  drops  into  the 
cutaway  at  J,  and  they  will  move  along  together  toward  K  and  be- 
come locked  there  upon  the  opposite  side  to  that  shown  by  the  cut, 
and  in  a  similar  manner.  When  the  limit  of  swing  is  reached  J 
returns  again  to  move  HI  back.  Thus  the  balance  staff  and  attached 
parts  are  free  from  contact  with  other  pieces  most  of  the  time,  since, 
when  locked  at  the  one  side  or  the  other,  the  escape-wheel  teeth  are 
locked  upon  the  pallets  as  shown  at  F,  where  the  angles  are  such 
as  to  draw  D  or  E  toward  A  during  the  lock  and  holding  the  pin 
H  away  from  the  edge  of  the  collar  J. 

When  H  moves  to  the  right,  the  tooth  at  JVis  unlocked  and  its 
inclined  end  passes  the  end  of  the  pallet  at  F,  exerting  a  pressure 
upon  it  and  imparting  an  impulse  to  the  pin  J.  Similar  action 
occurs  at  the  pallet  E  to  give  an  opposite  impulse  to  J  in  the  re- 
verse movement. 

The  pallets,  or  the  teeth,  or  both  should  be  rounded  by  the 
amount  of  the  angle  of  swing  of  B  as  shown,  to  prevent  the  corners 
of  D  or  N  from  scratching  each  other  while  in  motion. 

In  watches,  D  and  E  are  jewel  pallets  and  J"  is  a  roll  jewel. 

THE  DUPLEX  ESCAPEMENT. 

This  escapement   is  in   use  in   expensive  and  also  in  cheap 


FIG.  265. 

watches,  and  is  next  to  the  detached  lever  and  chronometer  for 
freedom  from  friction. 

Ay  Fig.  265,  is  the  escape-wheel,  B  the  balance  staff  with  a 


234  PRINCIPLES   OF   MECHANISM. 

longitudinal  groove  at  H  for  a  short  distance  to  admit  and  pass  the 
point  of  the  spur  H,  J,  etc.,  and  D  is  a  pallet  to  engage  the  impulse 
tooth  F. 

The  figure  represents  a  spur  just  escaping  the  groove  at  H,  and 
the  tooth  F  about  to  engage  with  D,  to  impart  to  it  an  impulse 
while  moving  from  D  to  the  point  of  escape  at  E.  Then  the  next 
spur,  J,  should  meet  the  staff  at  7,  leaving  a  trifle  of  clearance 
for  E  at  G  and  F.  The  pallet  moves  on,  to  the  limit  of  swing  of 
balance  and  staff,  and  returns,  passing  to  the  other  limit  of  swing, 
with  /rubbing  on  the  staff  at  7  and  skipping  the  groove  H.  Re- 
turning from  this  limit,  the  spur  J  drops  into  the  groove  H,  passes 
from  I  to  H  and  escapes,  etc.,  as  before,  the  escape-wheel  passing 
one  tooth  at  each  double  vibration  of  the  balance. 

The  friction  is  chiefly  the  rubbing  of  the  spur  on  the  staff  at  7, 
and  to  reduce  this  the  staff  should  be  made  as  small  as  consistent 
with  its  strength. 

This  is  properly  called  duplex,  because  of  the  two  points  of 
escape,  the  principal  one  at  GE  and  the  secondary  one  at  H. 

THE  CHRONOMETER  ESCAPEMENT. 

At  A,  Fig.  266,  is  the  escape-wheel,  B  the  balance  staff,  D  the 
pallet,  LGJKihe  detent  piece,  G  the  detent,  K  the  detent  spring 


FIG.  266. 

supporting  Z6r/and  forcing  it  against  the  banking  pin  N",  IJ  the 
feather  spring  bearing  against  the  pin  L  in  the  detent  piece,  and  7 
a  projection  on  the  staff  B  to  act  upon  the  feather  spring. 

In  action  the  projection  7  strikes  the  spring  IJ  resting  against 
7/,  and  forces  LGJ  away  from  the  pin  N,  and  the  detent  G  away 
from  under  the  tooth  at  G,  when  the  tooth  E  strikes  the  pallet  D, 


ESCAPEMENTS.  235 

imparting  an  impulse.  As  ^escapes  at  F,  a  tooth  is  caught  by  the 
detent  G.  The  balance  moves  on  to  the  limit  of  swing  and  returns 
with  D  just  clearing  F  and  E  and  with  the  projection  /  meeting  the 
very  light  feather  spring  ILJ  and  flexing  it  to  pass,  when  the 
spring  flies  back  against  the  pin  L.  On  reaching  the  opposite 
limit  of  swing,  the  balance  returns,  repeating  the  above  movements 
in  succession. 

To  reduce  the  friction  to  a  minimum,  the  staff  and  projection  at 
/should  be  as  small  as  admissible;  also  the  feather  spring  and  de- 
tent spring  very  delicate,  especially  the  former. 

In  watches  and  timepieces  subject  to  abrupt  displacement,  the 
detent  piece  should  be  balanced,  and  perhaps  pivoted,  to  prevent 
the  detent  from  being  jerked  off  its  tooth  at  G. 

Also  if  balanced  on  a  pivot  with  a  retaining  spring  to  hold  it  up 
to  the  pin  N,  it  should  be  as  light  as  possible  to  prevent  abrupt 
turns  in  the  plane  of  the  escapement  from  unlocking  the  detent. 


PART  III. 

BELT  GEARING. 

BELT  gearing  includes  all  members  in  machinery  concerned  in 
transmitting  motion  in  the  manner  of  a  belt  and  pulley,  such  as 
belts,  bands  or  chains,  pulleys  or  sprocket  wheels  for  continuous 
motion;  or  for  limited  motion,  where  a. rope,  strap,  or  chain  passes 
partly  or  several  times  around  sectoral  wheels,  to  which  the  ends 
are  made  fast,  as  in  the  windlass,  or  the  "  barrel  and  fusee "  of 
•chronometers,  English  watches,  etc. 


CHAPTER  XXI. 

VELOCITY-RATIO  VARYING. 

THE  GENERAL  CASE. 

THE  VELOCITY-BATIO. 

IN  Fig.  267  take  the  irregular  rounded  pieces  shown  as  fitted  to 
swing  about  axes  at  A  and  B,  with  a  flexible  connector  DE  passed 
over  and  beyond  those  points,  and  made  fast  to  the  rounded  bodies, 
so  as  to  admit  of  motion  to  some  extent  by  A  pulling  DE  and 
driving  B. 

In  a  small  displacement  where  D  moves  to  G,  E  will  move  to  J, 


FIG.  267. 

and  regarding  the  flexible  connector  as  in  extensible  in  length,  JK 

236 


BELT   GEARING.    ^^C/UIFORN\A**^        237 

will  equal  GH.  Also  the  triangle  HGD  is  similar  to  FAD,  and 
KJE  similar  to  IBE. 

If  V  and  v  be  the  angular  velocities  of  A  and  B,  we  have 

V.AD^DG,  v.BE=EJ, 

jDG_AD  EJ  _  BE 

HG  ~  AF'  JK  ^  ~m> 

,      .  V      DG   BE      AD  BI     I    BE 

and  velocity-ratio  =-=_/_=,_.  #0 ._._  -^ 

Bl       BC 
~  AF~  AC' 

which  shows  that  the  velocity-ratio  equals  the  inverse  ratio  of  the 
segments  of  the  line  of  centers  as  formed  by  the  intersection  of  the 
prolonged  line  of  centers  and  connector. 

The  line  of  the  connector  DE  is  here  the  line  of  action,  and  the 
velocity-ratio  is  the  same  as  found  in  all  previous  cases,  viz.,  equal 
the  inverse  ratio  of  the  segments  of  the  line  of  centers,  counting 
from  the  centers  of  motion  to  the  intersection  with  the  line  of 
action. 

The  same  is  true  if  the  connector  cuts  the  line  of  centers  be- 
tween A  and  B,  in  which  case  the  directions  of  motion  are  contrary, 
while  in  the  outside  intersection  they  concur. 

For  velocity-ratio  constant  the  point  C  must  be  stationary,  a 
condition  readily  secured  for  circular  pulleys;  also,  it  can  be  real- 
ized for  a  relatively  long  distance  between  centers  for  chain  and 
sprocket  wheels  even  with  the  latter  non-circular,  if  symmetrical 
and  with  axes  of  symmetry  at  right  angles,  and  for  velocity-ratio 
equal  to  1,  2,  4,  etc.  For  the  value  2,  the  larger  wheel  may  be 
nearly  a  square,  and  the  smaller  nearly  a  duangle.  But,  as  no  ad- 
vantage for  this  combination  over  circular  pulleys  is  conceivable, 
they  will  not  be  further  studied  here. 

But  non-circular  pulleys  for  velocity-ratio  variable  have  been 
used  with  advantage. 

LAW  OF  PERPENDICULARS  GIVEN  TO  FIND  THE  PULLEY. 

In  Fig.  268  take  A  for  the  axis  of  the  non-circular  pulley  and 
the  periphery  e,f,  g,  h,  to  be  found  when  the  line  of  action  Dg,  or 
of  the  connector,  has  a  known  perpendicular  distance,  Ag,  from  A 
for  each  1/8  turn  of  A. 


238 


PRINCIPLES   OF   MECHANISM. 


Lay  off  the  several  perpendiculars  as  radial  distances,  Aa,  Ab, 
Ac,  Ad,  Ae,  etc.,  and  draw  circles  or  otherwise  transfer  them  to  the 
several  radii,  marking  the  several  angles  through  which  A  turns  for 
the  corresponding  perpendiculars,  giving  points  e,  f,  y,  li,  i. 
Through  these  several  points  draw  the  perpendicular  line  of  the 
connector,  as  at  e  the  line  perpendicular  to  Ae,  at /the  perpendic- 
ular to  Af,  etc.,  to  represent  the  position  of  the  line  of  action  for 
the  respective  radii. 

Then  draw  the  curve  of  the  pulley  tangent  to  these  lines  of 


FIG.  268. 

action.  The  pulley  outline  does  not  pass  through  the  several 
points  h,  g,  f,  etc.,  but  puts  the  line  of  action  at  the  proper  per- 
pendicular distance  from  A. 

Some  examples  may  serve  to  illustrate. 

VELOCITY-RATIO  VARIABLE. 
DIRECTIONAL   RELATION   CONSTANT. 

1st.  Example  of  the  Equalizer  of  the  Gas-meter  Prover. 

The  meter  prover  consists  of  a  hollow  cylinder  of  some  20  cu. 
ft.  capacity,  to  be  raised  and  lowered  in  water  for  shifting  air  through 
any  gas  meter  for  testing  it.  The  more  the  cylinder  shell  is  lowered 
into  the  water,  the  greater  the  buoyancy,  this  being  counteracted 
by  a  weight  suspended  from  the  periphery  of  an  eccentric  pulley. 

To  determine  this  pulley,  let  DE,  Fig.  269,  represent  the  descent 
of  the  cylinder  shell  into  water,  the  surface  of  which  is  at  Ee,  equal 
steps  of  submergence  being  noted  at  points  d,  c,  b,  a. 


BELT-GEARING. 


239 


FIG.  269. 


This  cylinder  is  suspended  from  a  circular  pulley,  AF9  of  such 
size  as  to  make  a  3/4  turn  for  the 
range  DE,  upon  the  axis  of  which 
is  mounted  the  eccentric  pulley, 
e>  f>  9)  h>  i>  °t  a  like  3/4  turn,  to 
which  a  weight,  W,  is  suspended. 

By  trial  or  otherwise  find  the 
length  of  the  perpendicular  Ae, 
at  the  end  of  which  the  weight 
W  will  just  balance  DE  in  the 
highest  position.  Similarly  find 
the  perpendicular  distance,  Aa,  at 
which  W  will  just  balance  the 
submerged  cylinder  at  the  lowest 
position,  or  with  D  and  a  down  to 
the  water  surface,  Ee. 

Then  divide  ae  =  DE  into  the 
same  number  of  equal  parts  by 
points  b,  c,  d,  etc.,  as  there  are  equal 
angle  divisions  in  the  3/4  turn  e,  g,  i.  Then  make  Af  =  Ad,  Ag  = 
Ac,  Ah  =  Ab,  etc.,  and  through  these  points  draw  lines  of  action 
or  perpendiculars  to  Af,  Ag,  etc.  Now  draw  a  curve  as  shown 
tangent  to  all  these  last-named  perpendiculars  for  the  linear  profile 
of  the  eccentric  pulley  required. 

The  parts  being  made  and  mounted  as  thus  determined,  there 
should  be  perfect  balance  between  W  and  DE  in  any  position. 

2d.  Example  of  a  Draw-bridge  Equalizer. 

In  Mehan's  "Civil  Engineering "  is  illustrated  a  draw-bridge,  Fig. 
270,  having  a  rope  or  chain  with  one  end  attached  at  a  to  the  draw- 
bridge DE,  while  the  other  end  winds  upon  a  cylindric  drum  A, 
on  the  axis  of  which  is  an  eccentric  pulley,  Fig.  271,  upon  which 
winds  a  rope  or  chain,  to  the  lower  end  of  which  is  attached  a  weight. 

Here  the  weight  of  the  bridge  is  constant  but  causes  a  variable 
tension  upon  the  rope  Aa  as  the  bridge  is  opened,  which  is  to  be 
equalized  by  the  eccentric  pulley  and  weight.  Observing  that  the 
weight  of  the  bridge  moves  in  a  circle  a,  b,  c,  about  D,  the  rope 
tension  for  the  points  a,  b,  c,  etc.,  will  be 


tension  =  W^,     W-^, 


240 


PRINCIPLES   OF   MECHANISM. 


which  values  are  to  be  found  and  laid  off  on  the  line  Aj,  Fig.  271, 
giving  points  j,  0,  p,  q,  etc.,  for  perpendiculars  from  A  upon  the 
line  of  action.  Then  draw  an  involute  astu  to  the  circle  A.  Fisr. 

.7  o 

270,  and  take  the  distances  bs,  ct,  du,  etc.,  and  lay  off  on  the  circle 


m 


FIG.  270. 


FIG.  271. 


A  of  the  drum,  Fig.  271,  giving  angles  jAk,  j Al,  fAm,  etc.,  and 
draw  the  radial  lines  Aj,  Ak,  Al,  Am,  etc.  Now  lay  off  Ao  on  Ak, 
Ap  on  Al,  etc.,  and  draw  perpendiculars  to  these  radial  lines 
through  the  points  j,  k,  I,  etc.,  and  tangent  to  them  the  eccentric 
or  spiral  pulley  required,  as  shown. 


3d.  The  Barrel  and  Fusee. 

In  this  the  tension  on  the  cord  or  chain,  due  to  the  torsional 
action  of  the  spring  in  the  barrel,  is  to  be  determined  for  equal 

angular  positions  of  the  barrel 
through  the  entire  range  of  its 
motion.  Starting  with  the  spring 
slack  or  "run-down,"  find  the  ten- 
sion when  just  fairly  started  to 
wind  up,  as  for  the  chain  or  cord 
FIG.  272.  tangent  to  the  barrel  at  D,  Fig. 

272.  Then  pull  E  up  to  D  and  note  the  tension ;  then  F  to  D,  G 
to  D,  etc.,  noting  the  tension  for  each.  These  will  increase  as  the 
barrel  is  thus  wound  up  or  forced  around  against  the  action  of  the 
spring. 


BELT   GEARING.  241 

Through  the  center  of  the  fusee  at  B  draw  an  involute  Ba  to 
the  barrel  A.  Then  the  distance  from  D  to  the  involute  Ba,  fol- 
lowing the  cord  or  chain,  is  always  the  same,  whether  the  tangent 
point  is  at  D  or  farther  toward  B. 

Then  make  the  radial  distances  Ba,  Bb,  Be  inversely  as  the 
tensions  as  above  determined,  Ba  for  tension  at  D,  Bb  for  tension 
when  E  is  at  D,  Be  for  F  at  D,  etc. ;  and  make  the  distances  ab,  be, 
cd,  etc.,  equal  the  distances  DE,  EF,  FG,  etc.  Through  B  and 
these  points  b,  c,  d,  etc.,  copy  the  involute  Ba,  as  shown,  and  at 
these  points  draw  normals  to  the  involutes.  Then,  tangent  to  these 
normals,  draw  in  the  curve  of  the  fusee. 

In  this  construction,  the  smaller  the  angles  DAE  the  more 
accurate  will  be  the  result  until  the  limit  is  reached  where  the  in- 
accuracies of  graphical  work  predominate.  For  relatively  large 
angles,  DAE,  etc.,  it  will  be  advisable  to  make  the  distances  ab,  be, 
cd,  etc.,  which  are  chords  to  the  arcs,  equal  the  straight  dotted  line 
or  chord  DE.  Fig.  212  is  made  as  if  aD  were  to  act  by  compres- 
sion. To  change  the  figure  for  tension  in  aD,  place  b,  c,  d,  etc.,  to 
the  left  of  Ba. 

The  "  snail "  used  on  spinning  mules  is  like  a  double  fusee,  wind- 
ing a  cord  from  small  to  large  and  then  back  to  small  radii  again, 
thus  varying  the  velocity  of  the  cord  taken  up;  or  of  the  snail,  when 
the  cord  is  made  fast  at  the  remote  end. 

In  deep  mines,  where  heavy  wire  ropes  are  used  for  hoisting,  the 
winding  drums  may  be  made  conoidal  to  compensate  for  the  vary- 
ing load  due  to  the  varied  weight  of  rope  run  out  as  the  hoisting 
cage  is  let  down,  and  vice  versa.  If  the  length  of  rope  run  out  per 
revolution  were  constant,  the  drum  would  be  a  cone;  but  as  more 
rope  is  let  off  per  revolution  where  the  drum  is  larger,  its  shape  will 
be  a  concave  conoid. 

4th.  Non-Circular  Pulley  for  the  Rifling  Machine. 

There  was  in  use  in  the  rifle  factory  of  Windsor,  Vt.,  fifty  years 
ago,  a  rifling  machine  in  which  was  employed  a  belt  and  non-cir- 
cular pulley  connection,  for  the  purpose  of  imparting  to  the  rifle 
groove-cutting  tool,  held  in  a  rod,  an  approximately  uniform  motion 
forward  and  back  as  the  tool  and  rod  traversed  the  rifle  barreli, 
The  rod  was  driven  by  a  crank  and  pitman,  on  the  crank  shaft  of 
which  was  an  elliptic-shaped  pulley  similar  to  Fig.  275  connected 
by  belt  with  a  circular  pulley  above.  For  the  slow  motion  here 
employed  this  belt  and  pulley  combination  worked  satisfactorily. 


242 


PRINCIPLES   OF   MECHANISM 


To  determine  the  correct  form  of  non-circular  pulley  on  this 
crank  shaft  to  give  a  uniform  motion  to  the  slide,  consider  the  pit- 
g  man  of  infinite   length,  when  the 

projections  of  the  several  positions 
of  the  crank  pin  a,  h,  c,  g,  h,  i,  etc., 
upon  the  line  DE,  Fig.  273,  will 
divide  the  latter  into  equal  parts, 
E  r,  s,  t,  etc.,  representing  the  uni- 
form motion  of  the  slide,  as  if  the 
crank  pin  F  of  the  crank  BF 
were  moving  through  those  points. 
Drawing  Bg,  Bit,  etc.,  we  obtain 
the  angles  the  crank  must  pass 
FIG.  273.  in  equal  time.  On  these  lay  off 

the  respective  perpendiculars  br,  cs,  etc.,  giving  Bn,  Bm,  Bl,  etc. 
At  the  ends  of  these  draw  perpendiculars  as  in  Fig.  268.  These 
will  all  pass  through  the  point  a,  as  na,  ma,  la,  etc.,  and  drawing  in 

F G 


FIG.  274, 

the  pulley  curve  tangent  to  these  perpendiculars  gives  simply  the 
point  a,  which  shows  that  the  resulting  non-circular  pulley  is  merely 
a  flat  bar,  EF,  Fig.  274.     Though  this  seems  at  first  unreasonable, 
.it  is  correct,  since  the  belt  from  the  uniformly-moving  circular 
driving  pulley  A,  going  to  the  end  F  of  the  pulley  FE,  would  move 
^uniformly  in  the  direction  of  FG,  thus 
giving  the  crank  pin  at  F  the  motion  re- 
quired.    Hence  the  pulley  EF  should  be 
a  flat  bar  with  a  very  rapid  motion  when 
the  crank  pin  is  on  the  line  of   centers. 
This  pulley  would  work  with   a  suitable 
distance  between  centers  without  a  serious 
variation  of  tension  of  belt. 
At  the  dead  centers  of  the  crank,  the  jerk  of  speed  the  theory 
would  give  may  be  avoided  by  arbitarjly  widening  the  pulley  some- 
what as  in  Fig.  275.     For  moderate  speed,  as  for  a  rifling  machine, 


B 


FIG.  275. 


BELT    GEARING. 


243 


this  belt  and  pulley  combination  would  doubtless   be  thoroughly 
practical. 

DIRECTIONAL   RELATION   CHANGING    OR   CONSTANT. 

Prof.  Willis  has  devised  a  movement  answering  to  directional 
relation  and  velocity-ratio  varying,  called  "  cam-shaped  pulley  and 
tightener  pulley"  where  any  non-circular  pulley  may  be  mounted 
on  nny  center  and  mated  with  a  fixed  circular  pulley  and  a  tightener 
pulley  on  a  lever.  The  eccentric  pulley  should  be  the  driver  to 
avoid  slipping  of  belt.  (See  Willis,  page  201.) 

Example  of  a  Treadle. 

One  application  of  a  wrapping  connector  with  directional 
relation  changing  is  found  in  the  foot-lathe  treadle,  where  an 
eccentric  pulley,  E,  is  put  on  the  driving 
shaft,  and  a  centrally  mounted  pulley,  F, 
on  the  treadle  bar,  BD,  and  the  two  con- 
nected by  a  belt,  the  arrangement  serving 
to  avoid  making  a  crank  in  the  driving 
shaft,  as  in  Fig.  276.  The  action  is  evi- 
dently the  same  as  if  a  pitman  were  used 
to  connect  the  centers  E  and  F. 

It  is  not  necessary  that  the  pulley  E  be 
an  eccentric,  but  it  may  be  an  ellipse, 
with  A  in  one  focus,  or  a  flat  bar  clamped 
at  one  end  to  the  shaft  A. 

In  case  of  the  ellipse  of  the  same  length 
as  the  diameter  of  the  eccentric,  and  with  A  at  the  same  distance 
from  the  center,  the  full  stroke  of  the  treadle  bar  will  be  the  same 
for  both,  but  the  ellipse  will  lower  the  treadle  more  slowly  at  first 
and  more  rapidly  in  the  last  part  of  the  stroke.  If  A  be  at  the 
center  of  the  ellipse  and  the  latter  considerably  longer  than  the 
diameter  of  F,  the  treadle  will  have  two  short  double  strokes  per 
revolution  instead  of  one. 


FIG.  276. 


CHAPTEE  XXII. 

DIRECTIONAL  RELATION  CONSTANT! 

VELOCITY-KATIO    CONSTANT. 

HERE  the  pulleys  are  both  circular,  of  any  relative  dimensions, 
at  any  distance  apart,  and  with  belts  open  or  crossed  as  in  Fig.  277. 


FIG.  277. 


vrt 
The  velocity-ratio  is  always  equal  to  -      whether  the  belt  is 


open  or  crossed,  with  C  outside  or  between  centers.  When  G 
is  outside,  the  pulleys  turn  the  same  way,  and  when  between,  they 
turn  in  opposite  directions. 

The  exact  velocity-ratio  is  difficult  to  obtain,  for  two  reasons  : 
first,  the  thickness  of  the  belt  adds  somewhat  to  the  practical 
diameter  of  the  pulley,  and  relatively  more  for  the  small  than  large 
one;  and  second,  the  elastic  yielding  of  the  belt,  contracting  as  it 
goes  around  the  driving  pulley,  and  expanding  as  it  goes  around 
the  driven  pulley,  causing  a  "  slip  "  of  belt  which  increases  with 
the  driving  load  and  slackness  of  belt.  This  slip  keeps  the  pulleys 
bright. 

The  Belt. 

The  belt  may  be  a  strap  of  leather,  or  of  woven  stuff  sometimes 
filled  with  rubber;  or  of  leather  links  on  wire  joint  pins  built  up  to 

244 


CIRCULAR   BELT   GEARING. 


245 


any  desired  width;  or  a  strap  with  triangular  blocks  attached  to  run 
in  a  V-grooved  pulley;  or  a  round  belt  formed  of  thick  leather,  first 
cut  into  square  strips  and  then  rounded  by  drawing  through  a  die 
when  small,  as  for  driving  sewing  machines;  or  when  larger,  from 
-f  of  an  inch  diameter  up,  it  is  made  of  a  flat  strip  twisted  and 
pressed;  or  a  rope  of  hemp  or  wire.  All- but  the  flat  belts  run  in  V 
grooves. 

Chains  running  over  sprocket  wheels  now  constitute  a  most  im- 
portant connector. 

Retaining  the  Belt  on  the  Pulley. 

A  V  groove  of  sufficient  depth  will  naturally  retain  the  round 
belt  or  rope.  But  for  the  flat  belts,  the  pulley  is  made  "  high  cen- 
ter," since  a  flat  belt  has  a  tendency  to  climb  to  the  highest  part. 
This  is  due  to  the  edgewise  stiffness  of  the  belt,  giving  it  a  tendency 
to  climb  a  cone  of  moderate  taper. 

Thus, in  Fig.  278,  take  OAB  as  a  cone  cut  at  the  lower  element 
and  developed  by  rolling  out  to  DE  till  coincident  with  the  tangent 


FIG.  278. 

plane  ODE.  Placing  a  strip  of  belt,  DE,  upon  it,  we  see  that  it  runs 
off  the  developed  cone  surface  at  D  and  E,  though  at  the  middle 
point  it  is  parallel  to  the  base. 

Redeveloping  the  portion  of  conic  surface  CD,  and  the  belt 
with  it,  the  latter  is  seen  to  run  off  the  base  of  the  cone,  getting 
more  and  more  inclined  the  farther  it  goes. 


246 


PEINCIPLES   OF   MECHANISM. 


Hence  it  appears  that  a  flat  belt  running  on  a  pulley  with  a 
high  center  will  climb  to  the  highest  point. 

Advantage  may  be  taken  of  this  fact  to  carry 
a  belt,  not  very  wide,  at  quite  an  angle  to  the 
shafts,  to  avoid  an  obstacle,  A,  as  in  Fig.  279.  It 
is  here  only  necessary  to  make  the  pulleys  quite 
convex  and  a  little  larger  at  one  end. 

Grossed  Belts. 

In  a  crossed  belt  the  latter  must  have  a  twist 
of  180  degrees  between  pulleys  to  keep  the  same 
side  of  belt  to  pulley.     Thus  the  edge  elements 
of  belt  must  be  appreciably  longer  than  the  cen- 
tral ones,  by  the  ratio  of  the   hypothenuse  of  a 
triangle  to  its  base,  when  the  latter  is  the  length 
of  the  free  part  of  belt,  and  the  altitude  of  the         FIG.  279. 
triangle  the  half-circumference  of  a  circle  whose  diameter  is  the 
belt  width.      Hence  it  is  not  feasible  to  run  a  wide  belt  crossed  for 
a  comparatively  short  line  of  centers. 

Quarter-twist  Belts. 

A  quarter-twist  belt  is  subject  to  severe  strains,  due  to  distortion, 
as  well  as  the  crossed  belt,  more  for  relatively  large  pulleys  and 
less  for  small  ones,  as  illustrated  in  Fig.  280. 


FIG.  280. 


FIG.  281. 

Without  a  guide  pulley,  the  belt  will  immediately  run  off  the 
pulleys  if  the  latter  are  turned  backwards. 


CIRCULAR   BELT   GEARING.  247 

By  arranging  the  pulleys  as  in  Fig.  281,  and  adding  the  guide 
pulley  to  hold  the  guyed  part  of  the  belt  over  near  to  the  straight 
portion,  the  pulleys  may  run  either  way  without  throwing  the  belt 
off  and  with  less  strain  on  it.  The  guide  pulley,  however,  must  be 
placed  in  an  awkward  position,  its  central  transverse  plane  to  coin- 
cide with  the  center  lines  of  the  belt  above  and  below  that  pulley. 

Any  Position  of  Pulleys. 

The  driving  and  driven  pulleys  may  be  placed  in  any  possible 
relative  positions  and  connected  by  a  single  belt,  if  four  or  less 
guide  pulleys  be  provided,  four  for  the  comparatively  simple  case 
where  the  pulleys  are  on  parallel  shafts  and  not  in  the  same  plane. 
For  more  awkward  positions  it  will  usually  happen  that  tangents 
to  the  driving  and  driven  pulleys  can  be  made  to  intersect,  at  each 
point  of  which  a  single  guide  pulley  may  be  located,  with  meridian 
plane  coincident  with  the  plane  of  the  proper  tangents. 

Cone  Pulleys. 

Pulleys  in  a  series  of  steps,  as  employed  on  lathes  and  their 
countershafts,  by  which  the  speed  of  the  lathe  may  be  changed 
from  one  constant  speed  to  a  number  of  others  by  shifting  the  belt 
are  called  cone  pulleys  or  stepped  cone  pulleys.  Not  only  lathes 
but  a  large  variety  of  other  machines  require  these  cones,  so  that 
their  correct  construction  is  important. 

A  common  practice  is  to  make  these  steps  equal  on  one  of  the 
cones,  when  they  will  require  to  be  equal  on  the  other  for  a  crossed 
belt,  but  for  an  open  belt  not  for  uniform  tightness  of  belt. 

A  little  inquiry  will  satisfy  the  seeker  after  truth  that  the  cone 
diameters  should  be  such  as  to  place  the  steps  of  speed  in  geomet- 
rical progression,  that  is,  the  relation  of  any  one  speed  to  the  next 
should  be  the  same  as  that  to  the  next;  or  the  ratio  of  any  two  ad- 
jacent speeds  should  be  constant  throughout,  as  has  been  correctly 
maintained  by  Professor  John  E.  Sweet. 

For  instance,  in  a  "  back-geared  "  lathe,  the  ratio  of  speed  for 
the  first  two  sizes  should  be  the  same  as  for  the  last  two  sizes  of  the 
cone,  in  order  that  in  back  gear  the  ratio  of  speed  may  be  pre- 
served. Thus,  for  simple  ratios,  suppose  the  cones  give  the  geo- 
metric series  of  speeds  as  1,  2,  4, 8.  Then  the  back  gear  should 
continue  this  series  as  16,  32,  64,  128;  the  ratio  of  any  two  adjacent 
speeds  being  2.  That  due  to  going  into  the  back  gear  should  still 


248 


PRINCIPLES    OF    MECHANISM. 


be  the  same.  Otherwise,  suppose  that  the  series  of  speeds  were  as- 
sumed as  1,  2,  3,4;  when  the  back  gear  will  give  5,  10,  15,  20,  if 
the  gearing  is  such  as  to  give  the  first  figure,  5.  Then  the  ratios  of 
adjacent  speeds  will  give  the  series  of  ratios 

2,  1.5,  1.333,  1.25,  2,  1.5,  1.333, 

a  quite  irregular  set  of  figures.  Granting  that  they  are  correct  up 
through  the  cone  to  back  gear,  the  values  of  the  ratios  decreasing 
gradually  from  2  to  1.25.  Then,  instead  of  continuing  on  a  de- 
crease, there  is  a  sudden  jump  to  2  again,  after  which  a  second  de- 
cline— which  is  clearly  irrational. 

It  is  also  important  that  the  gearing  of  the  back  gear  be  proper 
to  this  rational  geometric  series  of  Prof.  Sweet  as  well  as  the  cones, 
that  the  ratio  of  adjacent  speeds  throughout  may  be  a  constant. 

A  convenient  way  of  realizing  in  a  drawing  this  geometric  series 
of  speeds  in  laying  out  a  pair  of  cone  pulleys  is  shown  in  Fig.  282, 
where  a  length  AD,  measured  by  some  scale,  represents  the  revolu- 


FIG.  282. 

tions  per  minute  of  the  countershaft;  Ba,  Bb,  Be,  etc.,  by  the  same 
scale,  the  speeds  of  the  lathe  or  other  cone;  BH,  BI,  BJ,  etc.,  the 
radii  of  the  countershaft  cone;  AP,  AQ,  AR,  etc.,  the  radii  of  the 
lathe  cone;  and  AB  the  distance  between  the  axes  of  the  cones. 
This  may  be  correctly  drawn  as  follows: 

From  B  draw  a  line  Bf  at  any  convenient  angle.  Draw  lines 
ah  and  Tib,  then  bg,  go,  cf,  etc.,  parallel  to  them  respectively,  when 
the  distances  Ba,  Bb,  Be,  etc.,  will  be  in  geometrical  progression 
and  will  represent  the  geometric  series  of  speeds  of  the  lathe  cones, 
Ba  being  the  slowest,  and  the  others  Bb,  Be,  etc.,  being  made  by 
trial  to  agree  with  the  series  required. 


CIRCULAR  BELT   GEARING.  249 

Draw  lines  Da,  Db,  DC,  etc.,  and  extend  them  to  intersect  the 
line  AB,  also  extended.  From  these  points  of  intersection  draw 
lines  TSH,  URI,  VKP,  etc.,  tangent  to  the  circle  GM,  drawing  the 
first  line  through  S  so  that  AS  may  measure  the  desired  smallest 
radius  of  the  lathe  cone;  or,  if  preferred,  the  line  VKP  may  be 
drawn  instead,  making  AP  the  desired  largest  radius.  When  the 
first  line  is  drawn,  strike  in,  tangent  to  it,  the  circle  GM,  from  a 
center  0  on  a  line  midway  between  AP  and  BH.  The  radius  of 
this  circle  is 

OM  =  —  nearly, 

and  is  to  be  calculated.  The  remaining  lines,  URI,  QJ,  etc.,  are 
drawn  tangent  to  this  circle,  giving  the  radii  of  the  lathe  cone  AP, 
AQ,  AR,  etc.,  and  radii  of  the  countershaft  cone  BH,  BI,  BJ, 
etc. ;  and  the  cones  may  be  drawn  as  shown. 

If  a  line  as  Db  does  not  intersect  A  B  on  the  drawing  board,  we 
must  make  Bb  :  BJ  :  :  AD  :  A  Q,  which  may  be  done  on  another 
diagram. 

The  diagram  shows  that 

AD:  Ba: :  AP  :  BK\ 

that  is,  the  speed  of  the  countershaft  is  to  the  slowest  speed  of  the 
lathe  cone  as  the  largest  radius  of  the  lathe  cone  is  to  the  smallest 
radius  of  the  counter  cone,  as  it  should  be  to  accord  with  the 
geometric  series. 

In  a  diagram  for  a  crossed  belt  the  lines  all  meet  in  one  point 
near  GE,  while  for  the  usual  case  of  open  belts  the  positions  of  the 
lines  must  be  "  doctored  "  by  being  drawn  tangent  to  the  circle  G M, 
which  shifting  of  the  lines,  however,  does  not  interfere  with  the 
velocity-ratio,  since  the  ratio  AS  to  BH  will  always  be  the  same 
when  the  line  is  drawn  through  the  point  T. 

To  determine  the  radius  OM,  let  AR  and  Ar,  Fig.  283,  stand 
for  the  corrections  of  the  radii  R  and  r  to  account  for  foreshorten- 
ing due  to  the  inclination  of  the  belt,  which  amount  call  Al.  Then 
as  TW  is  the  belt  inclination,  half  the  foreshortening  due  to  it  is 

k  W  =  XW  -  SN  =  1/2  Al  nearly,  and  =      (AR  +  Ar), 


250 
whence 


PRINCIPLES  OF  MECHANISM. 


_  27t(AR  -f  Ar) 


FIG.  283. 
Also,  by  Fig.  283, 

-g-  :  — g —  :  :  R  —  r  :  ^#  nearly, 

giving  (.£— r)s  =  Al .  AB  =  2nG2 .  AB. 

Again,  Fig.  282, 


ZGE  '.FM::R-riAB  nearly, 


whence 


R  -  r  =  ^ 


FM 


Eliminating  R  —  r,  we  get 


n 


OMiFM::FM'.ZGE  nearly, 


But 
giving 


the  required  radius. 

This  radius,  or  rather  the  position  of  the  point  0,  Fig.  282, 
differs  from  that  adopted  by  Mr.  0.  A.  Smith,  Trans.  A.  S.  M.  E.,. 


CIRCULAR   BELT    GEARING.  251 

Vol.  X.,  p.  269,  where  LO  =  0.3UAB.  The  latter  may  be  used 
in  constructing  Fig.  282,  if  preferred. 

In  the  extreme  example  of  AB  =  4:8",AS  =  1",  BH  —  17.284'% 
while  an  intermediate  pair  of  sizes  were  10",  the  difference  of  belt 
length,  as  given  by  Fig.  282,  and  the  carefully'  calculated  value, 
differed  by  only  about  0.2". 

Eeuleaux,  in  The  Constructor,  p.  189,  gives  an  interesting  dia- 
gram for  determining  cone  pulleys;  also  Prof.  J.  F.  Klein,  in 
Machine  Design,  diagrams  and  tables;  but  in  none  of  these  do  we 
find  the  very  important  consideration  of  the  geometrical  series  of 


In  practice  the  drawing  for  Fig.  282  may  take  an  uncomfortable 
length.  It  may  be  shortened  to  the  extent  that  the  angle  BTH 
does  not  much  exceed  30  or  40  degrees,  by  taking  for  the  actual 
distance  AB  a  fictitious  length  ab,  and  using  for  the  radius  OM 
the  value 


. 

nAB 

Actual  cones  have  been  employed  where  a  variation  of  speed  is 
desired  while  running,  the  belt  being  shifted  when  desired.  It  is 
usually  difficult  to  keep  the  belt  running  satisfactorily,  unless  the 
cones  are  unduly  long. 

The  Evans  friction  cones  is  a  good  arrangement  where  cones  are 
required,  in  which  the  cones  are  placed  with  large  and  small  ends 
opposite,  and  a  hoop  of  belting  around  one  cone,  somewhat  larger 
than  the  large  end,  arranged  with  a  guide,  which  hoop  makes  the 
bearing  point  between  the  cones.  By  shifting  the  guide  the  hoop 
is  shifted  and  the  speed  changed. 

Rope  Transmission,  Rope  Belting,  Etc. 

i  The  term  rope  transmission  was  first  applied  to  rope  belting 
used  for  transmitting  power  over  relatively  long  distances,  but  re- 
cently it  has  come  to  be  employed  as  a  substitute  for  leather  belting. 
For  transmission  wire  rope  has  largely  been  employed  without  a 
"  take-up  "  where  the  sag  of  the  rope  for  stretches  of  several  hun- 
dred feet  will  vary,  compensating  for  temperature  and  wear,  and 
still  maintain  sufficient  tension  for  service.  With  this  tension  and 
a  speed  of  the  rope  varying  from  3000  to  5000  ft.  a  minute,  a  large 


252  PRINCIPLES   OF   MECHANISM. 

amount  of  power  will  be  transmitted  with  only  a  half -turn  of  rope 
•over  the  pulley  for  frictional  contact. 

But  for  shorter  spans,  where  hemp  or  cotton  rope  is  used,  provi- 
sion is  made,  first,  for  "take-up"  for  maintaining  constant,  or  at 
least  sufficient,  tension;  and  second,  for  drawing  the  rope  as  it 
stretches.  In  cable  railways,  where  the  working  tension  of  the  rope 
is  carried  to  a  very  high  value  with  comparatively  low  speeds,  the 
take-up  and  increased  frictional  contact  with  the  driving  drum  are 
matters  of  the  utmost  practical  importance.  In  the  slower  speed 
arrangements  the  rope  is  to  be  consumed  by  a  higher  working  ten- 
sion, while  in  the  higher  speeds  the  rope  is  worn  out  by  a  high 
working  velocity,  causing  frequent  flexural  stresses  and  abraiding 
contacts  at  the  pulleys. 

In  Rope  Transmissions  there  are  two  leading  systems :  first,  where 
the  one  loop  of  rope  reaches  the  entire  stretch  of  the  system,  with 
a  driving  pulley  at  one  end  and  a  driven  one  at  the  other,  guide 
pulleys  of  less  diameter  being  introduced  at  points  between  to  carry 
the  rope,  the  whole  layout  of  rope  being  in  one  vertical  plane;  and 
second,  where,  one  loop  of  rope  passes  over  a  driving  and  driven 
pulley  only,  with  no  guide  pulleys,  beyond  which  is  a  second  loop 
of  rope,  likewise  mounted  and  driven  from  the  first,  and  beyond 
which  last  is  a  third  loop  of  rope  similarly  mounted  and  driven  from 
the  second,  and  so  on  for  as  many  bents  as  desired;  there  being  at 
each  intermediate  station  point  either  two  pulleys  made  fast  on  one 
shaft  or  a  double  pulley.  This  system  must  be  arranged  in  one  ver- 
tical plane,  but,  as  in  the  first,  may  pass  over  hills  and  valleys. 

The  driving  and  driven  pulleys  are  of  large  size;  first,  because  of 
the  high  speed  required,  and  second,  to  prevent  a  too  severe  flexing 
of  the  ropes.  The  bottoms  of  the  grooves  in  the  pulleys  are  fitted 
with  wood,  hard  rubber,  leather,  or  like  material,  to  increase  the 
friction  of  contact  with  the  rope. 

A  horizontal  angle  may  be  turned  in  either  of  these  systems  in 
at  least  two  ways:  first,  by  introducing  bevel  gears  at  any  station; 
second,  by  passing  a  rope  vertically  downward  or  upward  from  one 
pulley  and  immediately  upon  another  set  in  a  vertical  plane  making 
;any  desired  angle  of  deflection  with  the  first  plane. 

In  the  above,  wire  rope  is  usually  employed. 

In  Rope  Belting  several  half-turns  are  made  around  the  driving 
and  driven  pulleys  to  produce  the  necessary  frictional  contact.  This 
is  done  differently  for  long  than  for  short  stretches  between  power 
pulleys. 


ROPE   TRANSMISSION. 


First.  For  a  Short  Stretch,  Fig.  284  illustrates  a  mode  of  dupli- 
cating the  passes  of  rope  from  driving  to  driven  pulleys.  The  rope 
passes  from  the  take-up  or  tension  pulley  D  to  the  first  groove  of 


FIG.  284. 


B.  Passing  half  around  B  it  runs  to  the  first  groove  of  A.  Passing 
half  around  A  it  goes  over  to  the  second  groove  of  B  and  a  half 
around,  when  it  goes  to  the  second  groove  of  A,  and  so  on  for  all 
the  grooves  of  A  and  B  till  when  it  leaves  the  last  groove  of  B  it 
passes  to  and  half  around  the  take-up  pulley  />,  when  it  goes  to  the 
first  groove  of  B  again  and  repeats  the  circuit.  One  piece  of  rope 
spliced  into  a  single  loop  makes  the  entire  circuit. 

The  take-up  pulley  D  is  mounted  upon  a  slide,  and  has  a  weight 
to  produce  the  desired  tension  and  take  up  the  slack  due  to  varying 
length  of  rope. 

If  the  pulley  grooves  of  A  are  all  of  one  diameter,  likewise  of 
B  though  it  may  differ  from  that  of  A,  the  ropes  in  the  working 
tension  side  will  all  have  the  same  tension,  and  those  on  the  slack 
tension  side  will  be  in  equal  tension  with  each  other,  but  less  than 
that  on  the  working  tension  side,  but  not  if  the  grooves  differ  in 
diameter  on  either  one  of  the-  wheels. 

The  pulley  D  may  be  placed  outside  of  A  and  B,  as  if  the  rope 
passed  from  B  over  beyond  A  and  then  back  to  B  again,  but  with 
no  advantage. 

Second.  For  a  Longer  Stretch,  the  rope  belt  may  pass  over  a 
counter  pulley  at  each  end  of  the  system,  as  in  Fig.  285. 

Here  the  rope  between  A  and  B  is  in  higher  tension  than  in 
Fig.  284,  it  being  due  to  the  cumulative  action  of  all  the  half -turns 
of  rope  contact  with  the  driving  pulley. 

One  point  to  be  noticed  here  is  the  fact  that  the  rope  is  varying: 


254 


PRINCIPLES   OF   MECHANISM. 


its  tension,  or  seeks  to,  from  the  first  groove  of  A  to  the  last,  and 
consequently  varying  its  length.  On  this  account  the  pulley  A 
should  have  its  grooves  varied  in  diameter  from  one  end  to  the 
other,  and  likewise  for  the  counter  pulley  D.  The  same  should  be 
done  at  the  end  B.  The  counter  pulleys  D  and  E,  however,  may 
consist  of  separated  sheaves  loose  upon  the  axis  and  of  one  size. 
The  amount  the  grooves  of  A  and  B  are  to  vary  in  diameter  will 
depend  upon  the  unit  of  elastic  yielding  of  the  belt,  and  the  total 


FIG.  285. 

variation  of  tension  between  the  going  and  returning  sides.  The 
drop  in  tension  from  one  end  to  the  other  of  A  or  of  B  should  be 
uniform  per  groove,  and  likewise  the  diameter;  while  the  end 
diameters  should  differ  by  the  amount  an  equal  length  of  belt  will 
vary  as  due  to  the  total  variation  of  tension. 

The  counter  pulleys  D  and  E  may  both  be  stationary,  with  a 
single  turn  of  belt  going  over  to  a  single-groove  tension  pulley  F% 
Also,  D  and  E  may  be  geared  up  with  A  or  B,  and  become  power 
pulleys. 

Third.  For  Cable  Railways,  Haulage  Lines,  etc.,  the  pulley  and 
counter  pulley  A  and  D  are  connected  and  driven  by  power,  while 
at  B  there  is  only  a  single  sheave  for  returning  the  rope. 

Ingenious  compensating  devices  are  in  use  to  provide  for  the 
variation  in  rope  length  while  passing  over  the  series  of  grooves  on 
the  connected  driving  and  counter  pulleys. 

For  instance,  let  G  and  H,  Fig.  286,  represent  two  pulleys  of 
«qual  size  and  number  of  pulley  grooves,  tapered  from  H  toward 
G  to  partially  compensate  for  varied  rope  length  between  H  and 


.CHAIN   AND   SPROCKET. 


255 


G 


FIG.  286. 


G.  The  two  pulleys  are  loose  on  the  shaft  F,  but  driven  by  a 
bevel  gear  at  IL  and  JK,  mounted 
upon  axles  at  right  angles  to  the 
shaft  F,  and  made  fast  upon  it. 
These  gears  mesh  into  a  bevel  gear 
IJ  fast  in  the  pulley  H  at  IJ9  and 
into  a  bevel  gear  LK  fast  in  the 
pulley  G  at  LK.  Thus  mounted, 
if  H  is  turned  one  way,  G  will  be 
rotated  in  the  opposite  direction,  by 
reason  of  the  bevel  gearing  inter- 
posed. 

A  second  system  like  Fig.  286  may  be  prepared  and  the  two 
connected  by  gearing  to  serve  as  A  and  D  in  Fig.  285.  When 
rigged  with  rope  and  started  into  service  the  varied  diameter  of  H 
will  partly  compensate  for  varied  tension  and  length  across  H,  and 
likewise  across  G]  but  between  77 and  G  the  gearing  will  permit  a 
perfect  compensation,  by  revolving  slowly  in  opposite  directions. 
Then,  when  the  rope  is  working  under  a  greater  tension  than  the 
taper  of  pulleys  H  and  G  provides  for,  they  will  move  one  way  on 
the  shaft  F,  and  the  opposite  way  for  a  less  tension  than  the  taper 
of  pulleys  provides  for.  Thus  the  creep  of  rope  at  contact  with 
pulleys  is  reduced  by  the  splitting  of  the  pulleys  from  one  into 
the  two,  G  and  H. 


The  Chain  and  Sprocket  Wheel. 

A  wrapping  connector,  connecting  wheels  with  an  exact  velocity- 
ratio,  is  found  in  the  chain  so  formed  that  it  will  engage  the  teeth 
of  a  spur  or  sprocket  wheel,  as  in  the  familiar  example  found  on 
the  safety  bicycle,  for  a  connection  between  pedal  axis  and  driving 
wheel. 

The  chain  is  made  with  various  forms  of  link,  one  of  the  earlier 
ones  being  shown  in  Fig.  287,  where  the  links  are  punched  out  of 
sheet  metal  and  pinned  by  rivets,  while  a  more  modern  one  is 
shown  in  Fig.  288,  all  links  being  of  the  same  form,  sometimes 
cast  and  sometimes  drop-forged.  The  latter  possess  important 
advantages  in  having  the  axial  pin  solid  with  the  rest  of  the  link, 
and  in  having  each  piece  of  the  chain  like  every  other,  all  hooked 
together.  A  link,  turned  into  the  position  shown  at  Z>,  may  be 
removed  from  the  hook  and  more  or  less  links  added,  while  for 
working  positions  the  shoulders  a  and  b  prevent  unhooking.  It  is 


256 


PRINCIPLES   OF   MECHANISM. 


thus  easy  to  shorten  a  chain  when  worn  and  slack,  and  by  the 
amount  of  one  link  length;  while  for  Fig.  287  a  smith  is  required, 
and  the  least  that  can  be  removed  or  added  is  two  link  lengths. 


FIG.  287. 


'    Ml     I  H  ITTp 

*          "  1 1 


I     I    ! 

FIG.  288. 


The  Teeth  of  a  Sprocket  Wheel  engage  at  E,  F,  etc.,  or  H,  /,  Jr 
etc.,  Figs.  288  and  289,  as  the  case  may  be;  the  proper  forms  of 


FIG.  289. 


the  teeth  being  determined  as  shown  in  Fig.  289. 

Take  HGF  a  portion  of  the  chain  on  the  wheel,  while  at  F  it 
runs  off  in  the  tangent  FED. 

Now  as  FD  is  wound  upon  the  wheel,  the  center  point  E  de- 
scribes a  circle  arc  about  F  till  the  link  head  at  E  strikes  the  rim 
of  the  wheel,  when  the  links  EDI  will  swing  about  h  as  a  center 
till  D  strikes  the  rim  of  the  wheel,  and  so  on.  Then  we  will  find 
that  the  arc  from  a  to  the  straight  line  FJ  is  circular  about  h  as  a 


CHAIN   AND    SPROCKET.  257 

center;  from  .Fto  GFK  it  is  a  circle  arc  with  .Fas  a  center;  and 
from  K  to  L  it  is  a  circle  arc  about  G  as  a  center,  etc.  Then  a 
series  of  circle  arcs  ef,  fg,  parallel  to  ab,  Id  and  at  a  distance  dg 
from  it,  will  be  proper  for  a  sprocket  tooth  profile  for  the  circular 
wheel  A.  This  curve  copied  around  will  give  all  the  teeth. 

For  non-circular  sprocket  wheels,  as  in  the  cases  of  the  elliptic 
ones  used  on  some  bicycles,  the  sprocket-tooth  outlines,  to  follow 
theory  exactly,  should  each  be  determined  in  the  manner  of  Fig. 
289. 

Non-circular  sprocket  wheels  have  a  variation  of  velocity-ratio 
the  law  of  which  is  generally  simplest  when  one  of  the  pair  is  cir 
cular;  but  both  may  be  non-circular  and  in  any  ratio  of  sizes,  pro- 
vided the  wheels  both  have  symmetrical  axes  at  right  angles  to  each 
other,  observing  that  the  pair  of  wheels  should  be  so  far  apart  that 
the  inclination  of  the  chain  to  the  line  of  centers  does  not  become 
so  excessive  as  to  vary  the  tension  of  the  chain  unduly. 

Practical  Application  of  Chains. 

Chains  are  often  used  to  retain  a  mathematical  relation  between 
a  pair  of  axes  where  other  wrapping  connectors  would  not  answer. 

They  have  been  tried  for  the  transmission  of  power,  but  experi- 
ence shows  that  they  must  run  so  very  slowly  to  prevent  noise  and 
shocks  that  they  have. been  abandoned  for  this  purpose. 

Hooks  and  projections  have  been  formed,  on  the  links  to  which 
may  be  attached  conveyor  bars,  boards,  buckets,  scoops,  etc.,  when,, 
in  stretches  of  considerable  length,  sometimes  several  hundred  feet, 
running  vertically  or  on  inclined  tracks,  grain,  sand  bags,  refuse, 
etc.,  may  be  carried  in  a  continuous  current.  Thus  excavators 
have  been  operated,  and  coal-cutting  machines,  where  cutting  tools- 
are  made  fast  to  the  links. 


PART  IV. 

LINK- WORK. 

THIS  term  is  applied  to  such  machinery  as  consists  of  rods, 
cranks,  levers,  bars,  etc.,  jointed  together,  either  for  axes  parallel 
or  meeting,  or  crossing  and  not  meeting. 


CHAPTER   XXIII. 
THE   GENERAL   CASE. 

The  Velocity-Ratio. 

TAKE  A  and  B  as  fixed  centers  of  motion,  AD  the  driving  crank 
or  lever,  BE  the  driven  crank,  and  DE  the  connecting-rod  bar  or 
link.  As  AD  turns  about  its  center  A,  the  rod  DE  compels  BE  to 
turn  also;  Fig.  290. 

To  determine  the  velocity-ratio  compare  with  Fig.  267,  in  which 


AD,  DE,  and  BE  may  replace  the  lines  of  like  lettering  of  Fig.  290. 
Hence  for  the  latter  the 


V       BC 

velocity-ratio  =  —  =  -r-~. 
-  11        AC 


258 


THE   GENERAL   CASE.  259 

When  the  point  C  is  beyond  AB  the  cranks  turn  in  the  same  di- 
rection, and  in  opposite  directions  when  it  is  between. 


Peculiar  Features  of  Link-Work  Mechanism. 

Link-work  movements  are  the  most  arbitrary  in  law  of  motion 
of  all  the  combinations  of  mechanism,  and  in  many  cases  where  it 
would  otherwise  be  preferred,  must  be  abandoned  on  this  account. 
It  is  the  "lightest-running"  mechanism  known,  the  resistance 
being  due  to  the  slight  friction  of  comparatively  small  pins  in  bear- 
ing holes  well  lubricated,  and  with  comparatively  long  arms  with 
which  to  turn  those  pins.  Also,  a  pin  rarely  makes  more  than  one 
complete  turn  in  its  bearing  hole  in  a  complete  movement,  and 
usually  much  less;  while  in  the  corresponding  movement  of  a  cam 
motion  the  roller  (as  in  the  best  arrangement)  makes  from  6  to  12 
turns  on  its  pin,  and  even  this  is  not  so  prejudicial  as  regards  re- 
sistance as  the  rolling  of  the  roller  along  the  surface  of  the  cam 
groove. 

Link-work  is  consequently  much  more  durable  than  other  forms 
of  mechanism,  and  should  be  adopted  into  machinery  whenever  it 
can  be  in  preference  to  cam-work,  tooth-gearing,  etc. 

It  often  happens  that  the  principal  or  hardest  working  move- 
ments of  a  machine  may  be  link-work  while  cam  motions  adapted 
to  it  may  serve  for  other  movements,  and  give,  on  the  whole,  a 
more  satisfactory  machine  in  operation  than  where  all  are  cam 
movements. 

Here,  instead  of  assuming  a  law  of  motion  and  finding  the  link- 
work,  as  can  be  done  in  gearing,  cam  and  belt  movements,  etc.,  we 
must  examine  a  proposed  link-work  movement  complete,  when,  if 
found  unsuitable,  try  another,  etc.,  till  an  acceptable  one,  if  possi- 
ble, is  obtained. 

The  motion  sought  may  be  impossible,  while  an  approximation 
to  it  may  answer  by  conceding  some  unimportant  idiosyncrasy  of 
movement. 


I.  AXES  PARALLEL. 

Most  link-work  belongs  to  this  case  of  axes  parallel,  and  the 
treatment  must  largely  be  by  examples,  owing  to  peculiarities  and 
want  of  susceptibility  of  reduction  to  broad  and  extended  laws. 


260 


PRINCIPLES   OF   MECHANISM. 


FiG. 


Some  Examples  of  Link-work  Movements. 

Example   1. — To   illustrate:    An    excellent   needle-bar   motion 
for  a  shuttle  sewing-machine  is  obtained  by  link-work;  and  it  is, 
without  question,  in  one  case  at  least,  the  light- 
est-running shuttle  machine  yet  built,  the  whole 
machine  being  link-work  mechanism. 

The  needle-bar  motion  referred  to  is  shown 
in  Fig.  291,  used  in  several  different  sewing-rna- 
chines.  At  A  is  a  rocking  shaft  with  a  crank 
arm  AD  attached,  the  latter  reciprocating  be- 
tween the  angular  positions  AE  and  AF,  the 
connecting  link  DB  raising  and  lowering  the 
needle-bar  BJ. 

As  D  passes  the  line  A I  of  crank  and  link 
straightened,  the  bar  reaches  its  lowest  position. 
As  D  moves  on  to  its  limit  of  motion  at  F  the 
bar  is  raised  an  amount  GH,  forming  the  loop 
for  the  shuttle.  Now  it  is  not  necessary  for  the 
needle-bar  to  return  to  G,  but  on  tarrying  a 
moment  raised  to  //  for  the  shuttle  point  to 
fairly  enter  the  loop,  it  may  reasonably  enough  continue  on  its  up- 
ward journey.  By  a  cam  it  would  be  given  this  action,  but  by  the 
link-work  of  Fig.  291  it  must  return  to  the  lowest  point  again  be- 
fore making  its  ascent. 

Now  if  this  drop  by  the  amount  HG,  after  the  shuttle  enters 
its  loop,  is  considered  fatal  to  the  machine,  the  proposed  link  move- 
ment of  Fig.  291  must  be  abandoned;  otherwise  it  may  be  accepted, 
giving  a  much  lighter-running  and  more  durable  combination  than 
if  the  cam  and  pin  be  adopted,  or  such  a  movement  as  in  Fig.  252. 
Example  2. — Another  instance  is  found  in  the  Corliss  valve 
gear,  where  the  rocker  plate  swings  to  carry  the  pin  D  back  and 
forth  from  //  to  F.  The  valve  and 
stem  B  are  moved  by  the  lever  BE 
us  it  swings  from  BE  to  BG.  As  D 
moves  from  J  to  F  and  return  the 
valve  is  opened  and  closed,  and  from 
J  to  If  and  back  it  remains  closed, 
but  is  compelled  to  be  moving 
slightly,  as  due  to  the  double  versed-sine  DK.  If  this  movement 
DfTis  counted  out  of  the  question,  the  link-work  combination  of  Fig. 


FIG.  292. 


EXAMPLES    OF    LINK-WORK. 


261 


D 


292  must  be  rejected,  otherwise  accepted,  as  in  fact  it  has  been  by 
hundreds  of  builders  following  in  the  footsteps  of  Geo.  H.  Corliss. 

Example  3. — Another  example  is  given  in  Fig.  293,  where  the 
link-work  movement  rotates  a  shaft 
B,  by  the  arm  BE,  from  the  position 
BF  to  BE  and  back,  during  the  half- 
turn  of  the  main  shaft  that  drives  AD 
from  Alto  AD  and  back;  while  for  H 
the  remaining  half-turn  of  the  main 
shaft,  D  moves  from  I  to  H  and  back, 
during  which  BF  is  nearly  station- 
ary, its  simultaneous  movement  being 
from  F  to  G  and  back.  It  was  pre- 
ferred that  for  this  latter  BF  remain 
absolutely  stationary,  but  the  slight 
movement  FG  was  counted  less  ob-  FIG.  293. 

jectionable  than  cam-work,  as  compared  with  the  linkage  of  Fig. 
293,  so  that  the  latter  was  adopted. 

It  may  be  noted  that  the  slight  movement  FG  is  less  when  the 
two  arcs  HI  are  convex  the  same  way  than  when  convex  opposite 
ways,  and  more  so  as  A  is  nearer  to  B. 

Example  4. — As  another  example,  suppose  a  point  is  required 
to  move  from  E  to  F,  Fig.  294,  and  return  within  the  sixth  part 

of  a  turn  of  the  main  shaft  A.  A 
cam  may  be  used,  by  which  the  point 
is  thus  driven  and  then  allowed  to  re- 
main quiet  at  E  for  the  5/6  remaining 
turn.  But  if  there  is  no  particular 
objection  to  the  movement  of  the 
point  from  E  to  G  and  return,  instead 
of  remaining  stationary  at  E,  link- 
work  may  be  employed,  as  in  Fig.  294, 
where  A  is  the  main  shaft,  AD  the 
crank,  De  the  pitman,  and  eBG  = 
HBF  a  bell-crank  lever,  the  arrange- 
ment being  such  that  while  the  crank  pin  moves  from  c  to  b  the  re- 
quired sixth  part  'of  a  turn  of  A,  a  moves  to  #  and  back,  and  EtoF 
and  back,  thus  meeting  the  essential  conditions  of  the  movement. 

Path  and  Velocity  of  Various  Points. 

In  the  study  of  the  motion  of  linkages  it  is  sometimes  necessary 
to  determine  simultaneous  positions  throughout  the  movement  for 


Fm.  294. 


262  PRINCIPLES   OF    MECHANISM. 

all  the  joints,  beginning  with  the  driver  which,  for  uniform  motion, 
should  have  the  points  equidistant,  as  illustrated  in  Fig.  295.  Here 
the  driver  A  makes  its  circuit  with  uniform  motion  from  0,  1,  2, 
etc.,  to  8.  Corresponding  points  are  given  like  numbers  throughout. 
The  link  BE  swings  about  the  fixed  point  B,  and  the  end  F  of  the 


FIG.  295. 

link  DEF  describes  the  curve  F,  5,  6,  etc.,  and  returns  to  F,  all 
the  spaces  being  passed  in  equal  times  provided  A  moves  over  its 
numbered  spaces  in  equal  times. 

The  curve  described  by  any  point  of  the  link  DEF  may  be 
determined,  as  was  done  for  the  point  F  in  the  above  instance. 
Modifications  of  the  path  F,  5,  6,  etc.,  may  be  made  by  changing 
the  position  of  B,  or  length  BE,  or  angle  DEF,  etc. 

A  link  may  be  connected  at  F,  thus  extending  the  linkage  into 
a  train,  and  possibly  several  links  in  succession,  leading  through 
several  elementary  combinations. 

Sliding  Blocks  and  Links. 

Sliding  parts  are  sometimes  introduced,  either  for  realizing  the 
conditions  of  a  link  of  infinite  length,  or  for  simplifying  the 
mechanism,  a  notable  example  being  that  of  the  crosshead  of  the 
steam-engine. 

Sometimes  a  link  may  be  greatly  shortened  and  simplified  in 
construction,  as  shown  in  Fig.  296,  where  the  pin  J^7  moves  in  some 
curve  FE,  as  that  described  by  the  point  F  in  Fig.  295.  In  fact 
the  parts  shown  in  Fig.  296  may  be  employed  in  continuation  of 
those  of  Fig.  295  into  a  train,  F  moving  through  the  points  1, 2,  3,  4, 
etc.,  in  equal  times,  being  regarded  as  the  driver  for  the  elementary 
combination  of  Fig.  296.  The  point  G  is  the  fixed  fulcrum  pin  for 
the  bell  crank  HGD,  and  points  1,  2,  3,  etc.,  denoting  positions  for 
H9  may  be  found  as  corresponding  with  like  figures  of  the  curve 
FE,as  if  that  curve  were  the  one  through  F  of  Fig.  295,  or  again, 
as  corresponding  with  the  points  1,  2,  3,  etc.,  of  AD,  Fig.  295. 


EXAMPLES   IN   LINK-WORK. 


263 


The  block  FD  serves  as  a  short  link  connecting  the  straight 
pins  F  and  D,  upon  both  of  which  the  block  slides  to  accommodate 
the  versed  sines  of  the  curve  FE  and  of  the  circle  arc  about  G. 

As  these  pins  are  always  perpendicular  to  each  other,  the  block 
FD  is  a  simple  single  piece  with  the  two  holes  at  right  angles  to 


/T\ 


FIG.  296. 

each  other.  In  this  way  the  motion  of  DH  is  to  be  found  as  in 
sliding  contact,  and  on  this  account  the  movement  would  seem  to 
require  classification  there.  To  determine  the  velocity-ratio,  find 
the  center  of  curvature  B,  of  the  curve  FE  at  F,  and  draw  the  nor- 


mal  FC  to  the  pin  D,  and  the  velocity-ratio  = 
simple  when  the  curve  FE  is  a  circle. 


This  becomes 


CHAPTER  XXIV. 

THE  ROLLING  CURVE  OR  NON-CIRCULAR  WHEEL  EQUIVALENT 
FOR  LINK-WORK.— GABS  AND  PINS. 

FOB  every  elementary  combination  in  link-work  the  equivalent 
motion  can  be  obtained  by  wheels  in  rolling  contact. 


FIG.  297. 

Example  1. — In  Fig.  297  the  axes  A  and  B  are  connected  with 
link -work  as  shown,  AG  being  the  driving  crank,  MG  the  connect- 


THE   ROLLING   CURVE   EQUIVALENT   FOR  LINK-WORK.         265 

ing  rod,  and  BM  the  driven  crank.  Their  lengths  are  such  that 
they  will  all  come  to  coincide  with  the  line  of  centers  DA  CEB. 

To  find  a  rolling  curve  possessing  the  same  law  of  velocity-ratio 
as  the  link- work,  the  links  are  to  be  put  in  several  positions,  as 
AGMB,  etc.,  and  the  intersections  with  the  line  of  centers  found 
as  at  C  for  the  position  O M.  Now  the  velocity-ratio  in  link-work 
being  the  inverse  ratio  of  the  segments  AC  and  BC,  also  the  same 
for  rolling  wheels  in  contact  at  C,  it  follows  that  a  pair  of  wheels 
in  rolling  contact  at  C  have  the  same  velocity-ratio  as  the  links 
AGMB,  with  intersection  of  link  at  (7.  Therefore,  revolving  C  to 
T  in  BM  prolonged,  and  C  to  P  in  A  G  prolonged  toward  P,  we 
have  points  T  and  P  in  the  peripheries  of  the  rolling  wheels.  Other 
points,  Q,  D,  A  and  R,  8,  are  found  in  the  same  way,  when  with 
points  sufficient  the  curves  may  be  drawn  in  complete,  above  and 
below  the  line  of  centers,  giving  the  rolling-wheel  equivalents  of 
the  link-work. 

The  positions  of  these  curves  in  the  figure  are  proper  to  the 
crank  positions  of  AD  and  BE,  as  if  the  wheels  were  finished  and 


FIG.  298. 

made  fast  to  the  cranks  in  these  positions.  The  distance  B8  = 
EA,  where  the  rolling  suddenly  stops,  corresponding  with  the  limit 
of  the  crank  movement  where  the  crank  AD  and  link  take  posi- 
tions on  the  straight  line  A  0,  BO  being  the  extreme  position  of  the 
crank  BE  when  AG  +  GM  =  AO. 

In  Fig.  298  is  shown  as  separated  from  Fig.  297  the  link-work 
and  the  equivalent  wheels  due  to  the  intersections  of  the  link  DE, 


OF   THE 

UNIVERSITY 


266 


PRINCIPLES  OF  MECHANISM. 


with  the  line  of  centers.  Throughout  the  movement,  the  curves 
will  remain  continually  tangent  to  each  other  at  a  point  (7,  so  that 
the  point  of  tangency,  and  the  intersection  of  DE  with  AB  will 
remain  some  common  point  C  on  the  line  of  centers. 

In  Fig.  298  the  intersection  C  is  for  the  line  DE  with  the  line 
of  centers  AB.    If  BE  be  fixed  for  the  line  of  centers,  the  inter 


FIG.  300. 

section  will  be  that  of  the  lines  AD  and  BE,  both  prolonged,  the 
curves  for  which  are  LKN,  UJV,  Fig.  299;  and  Wa,  Xb,  and  YBZ, 
Fig.  300,  as  given  in  Fig.  297  and  shown  separately  in  Figs.  299 


THE   ROLLING   CURVE   EQUIVALENT   FOR   LINK-WORK.         267 

and  300,  all  extending  to  infinity,  due  to  FH  and  F'H'  becoming 
parallel  to  BE.  One  position  of  the  links  is  shown  at  BFHE,  Fig. 
299,  and  another  at  BF'H'E',  Fig.  300,  the  first  giving  an  inter- 
section at  /  and  the  second  at  I1 .  The  intersection  /  is  revolved  to 
the  positions  BF  and  EH,  Fig.  297,  for  points  in  the  curve,  while 
the  point  I'  is  revolved  to  BF'  and  EH'  extended  backwards,  be- 
cause this  part  of  the  curve  is  at  the  other  side  of  the  infinitely 
distant  points.  Thus  Fig.  297  embraces  the  three  sets  of  wheels 
of  Figs.  298,  299,  and  300. 

Example  2. — The  rolling-wheel  equivalent  of  the  crank  and  pit- 
man is  shown  in  Fig.  301. 

The  crosshead  A  as  driver  has  its  center  of  motion  at  an  infinite 
distance,  so  that  the  line  of  centers  is  BH  extended  to  infinity. 

Taking  the  crank  in  the  position  B  G  and  the  pitman  at  A  G, 
the  intersection  of  the  latter  with  the  line  of  centers  is  at  D  inAG 


A    J 


FIG    302. 


prolonged.  Revolving  D  to  the  crank  gives  F  for  one  point  in  the 
wheel  B.  Also  revolving  D  to  the  line  EA,  representing  the  crank 
line  from  infinity,  gives  the  horizontal  line  DE  and  the  point  E  in 
the  mating  wheel  A.  Proceeding  thus,  we  obtain  the  curves 
BFHB  and  JEHI  for  the  rolling- wheel  equivalent  for  the  crank 
and  pitman  motion  for  a  half- turn  of  the  crank.  The  proper 
position  of  the  crank  relative  to  the  wheel  BFHB  is  the  line  BH. 
The  curve  IHJ  is  supposed  to  be  made  fast  to  the  crosshead  and 
moving  with  it,  as  shown  in  Fig.  302. 


268 


PRINCIPLES   OF   MECHANISM. 


Example  3. — In  Fig.  303  we  have  an  example  of  two  cranks 
connected  by  a  connecting  rod  and  drag  link,  the  connecting  rod 


FIG.  303. 

having  a  fulcrum  pin  near  its  center  carried  on  a  swinging  link, 
the  lower  end  of  which  works  on  a  fixed  pin. 

For  this  somewhat  complicated  elementary  linkage  the  rolling 
wheels  are  worked  out  and  teeth  set  upon  them  for  a  pair  of  gear 
wheels,  the  same  being  combined  with  the  linkage  in  the  one 
movement  for  a  practical  illustration  of  the  equivalence  of  link- 
work  with  its  proper  rolling  wheels. 

The  gears  show  that  the  velocity-ratio  of  the  linkage  is  far  from 
constant,  though,  judged  from  the  link  movement  alone,  might  be 
taken  to  be  nearly  so.  By  measured  radii,  taking  the  driver  to  be 
moving  at  a  constant  rate,  the  ratio  of  the  fastest  to  the  slowest  for 
the  driven  wheel  is  over  3. 

Example  4.— An  interesting  linkage  is  found  in  the  Peaucelliers 


D 


FIG.  304. 


parallel  motion,  called  by   Prof.  Sylvester  the  "kite/'  shown  in 
Fig.  304  in  the  links  ABDE. 


THE    ROLLING   CURVE   EQUIVALENT   FOR   LINK-WORK.        269 

Making  AB  the  fixed  line  of  centers,  and  A  the  driver,  E 
describes  the  circle  EB  about  A,  and  D  describes  the  circle  through 
DG  about  B. 

Dotted  positions  of  the  links  are  shown  to  indicate  the  nature  of 
the  motion.  The  crank  AE,  it  is  readily  seen,  makes  two  revo- 
lutions to  one  of  BD ,  and  there  are  two  positions  for  dead  points, 
viz.,  when  the  joint  D  is  at  G,  and  at  180°  from  G. 

The  equivalent  rolling  wheels  are  readily  found,  one  point  for 
each  wheel  being  at  the  intersection  /  of  DE  prolonged,  J  being  its 
correct  position  for  A,  with  reference  to  the  crank  position  AE,  and 
also  the  correct  position  for  B  relative  to  the  crank  position  BD- 
The  curves  being  symmetrical,  we  may  revolve  J  about  A  to  I  and  J 
about  B  to  TTin  AE  and  BD  prolonged,  giving  points  /  and  Kin 
the  wheels  relative  to  the  cranks  for  the  positions  AB  and  BG,  as 
if  the  wheels  were  cut  in  material  and  made  fast  to  the  cranks,  the 
smaller  one  to  crank  AB,  and  the  larger  to  the  crank  BG. 

Thus  determining  a  sufficiency  of  points  and  locating  them  all 
with  respect  to  some  one  position  of  the  cranks,  we  obtain  the  out- 
line of  the  wheels  shown. 

The  wheels  fully  worked  out  are  shown  together  with  the  link- 
work  in  Fig.  305,  where  the  B  wheel  is  seen  to  make  one  complete 
circuit  of  the  rirn  with  a  side  offset  in 
it  suitable  for  the  mating  wheel,  and 
where  the  A  wheel  makes  two  com- 
plete convolutions,  one  being  out  of 
the  plane  of  the  other  to  prevent  in- 
terference and  mating  respectively 
with  the  offset  parts  of  B. 

The  wheels  in  this  construction 
work  by  rolling  contact  of  pitch  lines, 
except  at  the  dead  points,  where  half- 
teeth  are  introduced  as  shown. 

Example  5. — The  case  of  two  equal 
cranks  revolving  in  opposite  directions  FlG-  305 

and  a  connecting  link  of  the  same  length  as  the  line  of  centers,  but 
longer  than  the  crank,  is  shown  in  Fig.  133,  accompanied  with  the 
equivalent  wheels,  the  latter  being  the  rolling  ellipses.  Hence  the 
law  of  velocity-ratio  for  this  linkage  is  the  same  as  that  of  a  pair  of 
rolling  ellipses,  each  with  the  axis  at  a  focus. 

Example  6. — The  same  linkage  as  in  Example  5,  except  that  a 
shorter  instead  of  a  longer  member  is  fixed  to  serve  as  a  line  of  cen- 


270  PRINCIPLES   OF    MECHANISM. 

ters,  is  shown  in  Fig.  306,  where  the  equivalent  rolling  curves  are 
hyperbolas  instead  of  ellipses.  The  hyperbolic  wheel  ECG  is  fixed 
upon  the  arm  BE,  while  FCH  is  fixed  on  the  arm  AD. 

For  the  positions  A  /and  BJ  of  the  arms  these  hyperbolas  are 
tangent  at  the  point  K  of  intersection  of  //  with  the  line  of  centers 


FIG.  306. 


AB,  both  prolonged.  The  points  H  and  G  in  the  hyperbolas  are 
obtained  by  revolving  Kio  the  radii  Aland  BG.  These  hyperbolas 
will  roll  in  mutual  contact  from  where  the  link  // is  parallel  to  the 
line  of  centers,  through  180  degrees,  to  where  it  is  again  parallel  to 
the  line  of  centers,  when  another  pair  of  hyperbolas,  shown  dotted, 
will  come  into  rolling  contact  for  the  remaining  180  degrees. 

The  dotted  hyperbolic  half-wheel  LM  is  to  be  fixed  upon  the 
arm  AD,  as  well  as  the  half-wheel  FH,  while  the  remaining  two 
half-wheels,  EG  and  NP,  are  to  be  fixed  upon  the  arm  BE,  and  at 
a  distance  between  vertices  equal  AB. 

The  points  B  and  D  are  foci  to  the  hyperbolas  EG  and  FH. 
Also  A  and  E  are  focal  points  for  the  dotted  hyperbolas.  Thus  A 
and  D  are  focal  points  for  the  hyperbolas  FH  and  LM,  the  asymp- 
totes for  which  intersect  each  other,  and  the  line  AD  at  its  middle 
point. 

A  pair  of  these  hyperbolic  wheels  mounted  for  non-circular 
wheel  pitch  lines  is  shown  in  Fig.  49,  where  AB,  as  in  Fig.  306,  is 
the  constant  difference  of  lines  drawn  from  all  points  C  to  the  focal 
points. 

The  elliptic  wheel  of  Example  5  and  Fig.  133  may  coexist  along 
with  the  hyperbolic  wheels  of  Fig.  306,  except  that  one  of  the 
ellipses  would  be  fixed  with  A  and  B  as  focal  points,  while  the 


THE    ROLLING    CURVE    EQUIVALENT    FOR    LINK-WORK.         271 

other  would  be  carried  on  the  link  //,  their  point  of  tangency 
being  constantly  the  intersecting  point  of  A I  with  BJ,  as  shown  in 
Fig.  307,  while  the  hyperbolas  are  tangent  where  AB  and  //inter- 
sect at  D. 

A  most  interesting  and  instructive  figure  in  this  connection  is 
brought  out  by  aid  of  his  theory  of  Centroids  by  Reuleaux,  and 
'given  in  his  Kinematics  of  Machinery  by  Kennedy,  p.  194,  much  as 
shown  in  Fig.  307,  where  the  linkage,  the  rolling  ellipses,  and  hy- 
perbolas are  all  presented  in  one  view. 

Thus  with  AB  the  line  of  centers,  //  may  be  a  link  connecting 

Hi 


FIG.  307. 

the  focus  /of  PJVwith  the  focus  /  of  HD,  as  has  been  admirably 
shown  by  Geo.  B.  Grant  in  his  handbook  on  Teeth  of  Gears,  Second 
Edition. 

Examining  Figs.  133  and  307,  we  find  that  for  the  ellipses  both 
the  link  and  line  of  centers  in  length  equal  the  sum  of  the  dis- 
tances from  the  point  of  tangency  of  the  rolling  ellipses  to  the  pair 
of  foci  A  and  B  on  the  opposite  sides  of  the  center  of  the  ellipse. 
In  the  hyperbolas  the  link  or  line  of  centers  equals  the  difference  of 
these  distances  from  the  point  of  tangency. 

Referring  these  curves  to  their  conic  sections,  we  find  that  the 
parabola  lies  intermediate,  and  is  like  the  ellipse  of  infinite  length, 
or  the  hyperbola  of  infinite  distance  between  foci. 

Example  7. — Therefore  we  may  expect  that  the  link  connecting 
foci  of  the  parabolas  would  be  of  infinite  length,  as  shown  by  Geo. 
B.  Grant  in  his  Teeth  of  Gears,  Second  Edition,  and  may  be  real- 
ized in  the  manner  of  Fig.  308,  where  AB  is  the  fixed  line  of  cen- 
ters, C  the  point  of  contact,  A  and  D  foci,  DE  a  link  sliding  on  a 
straight  guide,  mounted  on  the  parabola  A,  perpendicular  to  its 


272 


PRINCIPLES   OF  MECHANISM. 


geometric  axis  and  giving  to  D  the  same  motion  as  if  DE  extended 
to  infinity  and  were  pivoted  there  to  the  axis  or  opposite  focus  of 
A.  The  parabola  D  slides  on  a  guide  FG  perpendicular  to  the 


E 


FIG.  308. 

geometric  axis  of  D.     The  point  of  contact  C  is  at  the  intersecting 
point  of  the  link  and  line  of  centers. 

In  Fig.  309  we  realize  this  link-work  movement  without  the 
parabolas.     The  gabs  and  pins  will  be  explained  later. 

Example  8. — The  above  examples  are  all  for  curve  equivalents 


FIG.  309. 

that  are  symmetrical;  Example  2,  apparently  not, becomes  so  when 
the  remaining  half-revolution  is  provided  for. 


THE   ROLLING  CURVE   EQUIVALENT   FOR   LINK-WORK.         273 

As  an  example  entirely  wanting  in  symmetry,  take  the  link-work 
of  Fig.  293,  the  rolling-curve  equivalents  for  which  are  shown  in 
Fig.  310. 

The  links  are  shown  in  the  position  such  as  places  the  joint  E 


FIG.  310. 

on  the  line  of  centers  AB,  and  in  the  proper  relation  to  the  curves 
for  mounting  them  upon  the  crank  arms,  IL  and  OP  on  AE,  and 
<7^and  MN  on  £D. 

To  find  a  pair  of  points  in  the  curves  JK  and  IL:  Place  AE  in 
the  position  Ab,  when  the  link  ED  will  follow  to  the  position  be, 
giving  the  intersection  C  with  the  line  of  centers.  Then,  as  before, 
revolve  the  point  C  to  the  arm  Ab,  giving  the  point  e.  Now  if 
the  curves  were  symmetrical  this  point  could  be  retained  as  a  point 
in  proper  location  of  the  curve.  But  here  e  must  be  placed  at/' 
symmetrically  on  the  opposite  side  of  the  line  of  centers,  so  that 
when  the  arm  moves  from  E  to  b  this  point  will  move  up  to  the 
line  of  centers.  Also,  the  mating  point  a,  is  to  be  placed  at  i,  a 
being  on  a  line  Ba,  where  the  angle  CBa  =  DBc.  In  like  manner 
find  other  points. 

These  curves  reach  infinity  when  the  link  DE  becomes  parallel 
to  the  line  of  centers,  following  which  the  curves  MN  and  OP 
come  in  from  infinity.  The  point  of  OP  at  A  is  in  contact  with 
its  mate  when  the  link  DE  is  in  a  line  running  through  A ;  the 
portion  A  0  answering  to  the  slight  return  of  D  from  its  extreme 
point,  as  E  moves  from  the  straight  line  DA  toward  the  extreme 
point  F.  For  the  other  extreme,  Gf,  of  movement  of  E,  the  points 
/  and  /  are  in  contact  on  the  line  of  centers.  The  movement  of 
D  is  slight  for  the  movement  of  E  from  R  to  F,  while  for  RG 
the  crank  B  makes  over  a  90-degree  movement.  The  point  R  is 
about  midway  between  F  and  G,  so  that  D  is  comparatively  quiet 
for  half  the  time. 


274 


PRINCIPLES    OF    MECHANISM. 


Dead  Points  in  Link- Work. 

A  dead  point  or  dead  center,  in  link-work  mechanism,  is  a  point, 
or  set  of  points,  or  positions  of  the  links,  at  which,  if  certain  of  the 
links  in  combination  be  made  driver,  the  linkage  will  be  found 
positively  locked.  Thus  when  the  crank  and  pitman  are  in  line 
the  crank  cannot  be  started  into  motion  by  force  applied  to  the 
crosshead.  Dead  points  must  always  be  provided  against,  either  by 
inertia,  by  springs,  or  by  extraneous  attachments. 

In  the  steam  engine  the  inertia  of  the  fly-wheel  serves.  In 
some  reverse  motions  in  machines  springs  are  employed.  In  start- 
ing inertia  is  dead  also;  and  so  the  single-acting  steam  engine  must 
not  stop  on  the  dead  center.  In  locomotives,  two  sets  of  cranks 


FIG.  311. 

are  placed  at  right  angles,  so  that  one  is  at  its  best  advantage  when 
the  other  is  on  its  dead  center.  Any  angle  will  serve  with  a  degree 
of  efficiency,  as  illustrated  in  Fig.  311  of  Boehm's  movement,  where 
an  extra  link  is  added  to  destroy  the  dead  point.  Several  links 
may  be  added. 

Another  arrangement  is  shown  in  Fig.  312,  where  an  extra  crank 


FIG.  312. 


D,  of  radius  equal  that  of  A  or  B  is  added,  together  with  the 
side  extension  AD  and  DB  to  the  main  link  AB.  The  velocity- 
ratio  is  constant  in  the  locomotive  and  in  Figs.  311  and  312,  the 


DEAD   POINTS   IN   LINK-WORK.  275 

latter  being  obtained  from  Figs.  133  and  307  by  cutting  away  the 
wheels  and  making  the  link  parallel  to  the  axis. 

In  the  steam  engine  two  pitman  links  at  about  right  angles 
are  sometimes  connected  on  the  same  crank  pin,  thus  realizing  con- 
ditions equivalent  to  the  use  of  two  cranks  with  parallel  rods. 

Prof.  Reuleaux  has  introduced  a  gab  and  pin  at  the  points  of 
the  equivalent  rolling  curve  where  they  are  tangent  to  each  other, 
when  the  linkage  is  at  the  dead  centers.  Thus  in  Fig.  306  a  gab 
and  pin  is  to  be  placed  at  C,  where  the  hyperbolas  are  tangent,  one 
on  the  link  AD  and  the  mate  on  the  link  BE. 

In  case  a  longer  link  is  fixed  for  the  line  of  centers  the  gab  and 
pin  attached  to  it  become  fixed  also,  as  in  Fig.  313. 

Also,  a  gab  and  pin  may  be  placed  where  the  ellipses  become 
tangent  for  the  position  of  the  line  of  centers,  as  shown  in  Fig.  314. 


FIG.  318.  Pro.  314. 

Here  also  a  gab  and  pin  become  fixed  when  a  shorter  link  is  fixed 
for  the  line  of  centers,  as  shown  in  Fig.  315. 

In  Fig.  133  the  gab  and  pin  are  in  the  form  of  a  gear  tooth  and 
a  space  for  the  same,  and  represent  the  same  case  as  Fig.  314,  ex- 
cept that  the  elliptic  wheel  equivalents  of  the  link- work  are  present 
and  mounted  with  their  respective  links. 

In  Fig.  305  a  half  gear  tooth  and  half -space  are  made  to  serve 
for  gab  and  pin  to  carry  over  the  dead  center. 

In  Fig.  308  a  gab  and  pin  may  be  placed  either  at  the  vertices 
of  the  parabolas,  or  upon  the  links,  as  shown  in  Fig.  316,  one  set 
being  sufficient.  The  dead  center  occurs  when  the  swinging  piece 
is  vertical,  since  at  that  position  the  parts  DE,  FG,  Fig.  308,  may 
move  up  or  down  without  control.  One  set  of  gabs  and  pins 
between  A  and  D  will  prevent  this. 


276 


PRINCIPLES   OF    MECHANISM. 


In  Fig.  297  the  curves  show  that  a  gab  and  pin  may  be  placed 
at  O  as  indicated  by  either  system  of  curves.  In  Fig.  298  the  pin, 
for  instance,  may  be  placed  on  AD  extended  downward  from  A  to 
the  curve  at  F9  while  the  gab  may  be  placed  on  the  line  of  BE  ex- 
tended to  the  curve  at  G.  Thus  the  gab  and  pin  are  placed  on  the 


FIG.  315. 


FIG.  316. 


crank  arms  AD  and  BE,  so  that  both  are  in  motion.  But  a  pair 
may  also  be  placed  on  the  longer  members  as  at  H  and  /,  where  the 
latter  is  fixed  on  AB.  Thus  two  pairs  of  gabs  and  pins  are  avail- 
able in  this  linkage,  either  of  which  may  be  adopted,  whichever 
member,  as  AB,  AD,  DE,  or  BE,  is  the  fixed  one,  and  in  any  case 
one,  a  gab  or  pin,  may  become  fixed,  as  at  /. 

Figs.  298,  299,  and  300  show  the  possible  arrangement  of  all  the 
gabs  and  pins  for  this  linkage. 

The  linkage  of  Fig.  307  shows  eight  gabs  and  pins,  any  two  pairs 
of  which  may  be  adopted  in  a  particular  case— sometimes  all  in 
motion,  and  sometimes  two  being  fixed,  as  illustrated  in  Figs.  313 
to  315. 

In  Fig.  316  four  of  the  gabs  and  pins  are  at  infinity;  while  of 
the  four  shown,  one  is  fixed. 

Path  of  the  Gab  and  Pin. 

In  Fig.  317  is  shown  a  link  and  crank  connection  recently 
adopted  with  success  into  a  machine  to  transfer  motion  from  one 
ehaft  to  another  parallel  to  it,  the  gabs  and  pins  being  employed. 
The  connecting  link  is  raised  from  its  position  a  distance  FG  =  DE 
to  uncover  the  gabs  at  a  and  I. 

The  curves  described  by  the  centers  of  the  pins  a  and  b  are 


DEAD    POINTS   IN    LINK-WORK. 


277 


drawn  in,  to  show  how  the  gabs  must  widen  in  amount  of  opening. 
The  pins  a  and  b  are  placed  one-sided,  for  the  reason  that,  if  placed 
central,  they  would  interfere  with  the  gabs.  But  an  examination 
of  the  path  curves  for  «,  b,  and  c  shows  that  these  curves  are  cusp- 


FIG.  317. 

shaped  at  the  gab  positions,  and  that  the  central  one,  c,  does  not 
differ  materially  from  those  at  the  sides  at  a  and  b. 

As  to  the  advantage  or  disadvantage  of  placing  the  gabs  some- 
what off  of  the  line  of  centers,  very  little  difference  will  be  noticed. 

The  position  c  is  right  for  the  rolling  ellipses  of  Fig.  133,  but 
here  the  eccentricity  is  so  slight  that  interference  occurs. 

The  path  curves  for  «,  J,  and  c  were  traced  by  means  of  a  paper 
templet  on  the  drawing  board,  the  templet  being  so  cut  that  the 
edge  fitted  at  the  points  F,  a  and  D,  and  had  a  mark  at  each.  This 
templet,  placed  in  the  various  positions  of  F  and  D  in  the  circles, 
and  a  noted  for  each,  gave  points  through  which  the  curves 
were  traced. 

An  example  of  a  linkage  nearly  like  that  of  Fig.  299  in  applica- 
tion, is  given  in  Fig.  318,  which  picks  up  the  staple  at  the  lower 


FIG.  318. 


end  of  the  machine  and  delivers  it  at  the  upper  end,  thus  represent- 
ing the  handling  of  rods,  screws,  etc.,  in  the  manufacture  of  those 
articles. 


278 


PRINCIPLES    OF    MECHANISM. 


In  Fig.  319  is  given  a  model  of  link-work  serving  to  prove  that 
vibratory  motion  may  be  multiplied  thereby.  Thus  the  pointer  is 
given  four  movements,  to  one  of  the  first  vibrating  bar  of  the  series. 


FIG.  319. 


CHAPTER  XXV. 
LINK-WORK— (CONTINUED). 

H    AXES  MEETING. 

THIS  is  sometimes  called  conic  link-work,  or  solid  link-work,  the 
principal  essential  consisting  in  bringing  all  the  axial  lines  of  shafts 


Fib.  320. 

and  pins  to  a  common  point,  0,  as  in  Fig.  320.  Thus  we  may  have 
equal  cranks,  as  in  the  counterpart  for  parallel  axes  of  Figs.  306  and 
307;  or  unequal  cranks,  as  in  Fig.  299. 

To  realize  the  principles  of  equal  cranks,  it  is  only  necessary  to 
gives  the  cones  AOD  and  BOB  an  equal  slant. 

Any  of  the  examples  under  axes  parallel  may  be  carried  into 
conic  link-work,  even  to  the  extent  of  continued  trains. 

The  Velocity-Ratio  in  Conic  Link- Work. 

It  will  be  advisable,  if  not  necessary,  to  refer  problems  in  conic 
link-work  to  spherical  surfaces  normal  to  all  the  axes,  or  which 
have  Of  Fig.  320,  for  the  center  of  the  sphere. 

In  practice,  avoiding  spherical  trigonometry,  the  velocity-ratio 

279 


280 


PRINCIPLES   OF   MECHANISM. 


may  be  most  readily  determined  by  preparing  the  proper  spherical 

surface  in  wood  or  other  material, 
and  using  it  for  the  drawing  board. 
Take  Fig.  321  to  represent  this 
drawing  board  and  drawing. 

A  and  B  are  the  points  where  the 
axes  A  and  B  pierce  this  spherical 
surface;  D  and  E  where  the  crank- 
pin  axes  pierce  the  sphere;  AB  is 
the  line  of  centers  and  DE  the  line 
of  the  link,  both  being  parts  of  great 
circles,  and  constant  in  length  on  the 
sphere. 

Suppose  A  makes  a  slight  turn, 
moving  E  to  k,  then  D  will  move  to 
I  so  far  that  the  projections  of  Dl 
and  Ek  upon  DE  will  equal  each 
other  and  give  Do  equal  to  En,  equal 
to  pr,  equal  to  st',  since  the  spherical 

connecting  link  DE  remains  of  constant  length,  s  and  p  being 
points  noted  thereon  where  Ap  and  Bs  are  perpendicular  to  DE. 
Then  the  angle  sBt  measures  the  angular  displacement  of  axis  B, 
corresponding  with  that  of  pAr  for  A.  But  sB  is  a  circle  arc  on 
the  sphere,  also  Ap.  Eevolving  s  to  u  and  p  to  v  and  transferring 
the  arc  Bu  and  Av  to  bd  and  ae  in  the  great-circle  section  abO,  we 
readily  obtain  the  perpendiculars  eg  and  hd,  the  inverse  ratio  of 
which  is  the  velocity-ratio,  because 


'o 

FIG.  321. 


hd  X  angle  sBt  —  st  =  pr  =  eg  x  angle  pAr, 


giving 


velocity-ratio  = 


_  angle  sBt eg 


angle  pAr      hd 


Hence  in  practice,  for  any  position  of  links  as  AEDB,  draw  the 
perpendiculars  to  the  links  Ap  and  Bs,  which  arcs  transfer  to  ae 
and  bd.  Then  the  velocity-ratio  equals  the  inverse  ratio  of  eg 
and  dh. 

The  Rolling- Wheel  Equivalent  of  Conic  Link- Work. 

In  Fig.  321  draw  parallels  ef  and  df  to  the  axes  Oa  and  Ob,  and 
through /draw  Oc.  Then  ac  and  be  will  be  the  spherical  arc  radii 


II.    AXES    MEETING.  281 

for  the  spherical  rolling-wheel  equivalents  of  the  link-work  of  Fig. 
321,  for  contact  occurring  when  the  links  are  in  the  positions 
shown.  When  the  links  are  located  for  drawing  the  wheel,  as  for 
instance  AE  on  the  line  of  centers,  their  radii  may  be  laid  off  in 
place.  Likewise  for  other  radii,  until  a  sufficient  number  of  points 
.are  determined  for  drawing  in  the  wheel. 

The  Dead  Center  and  Gab  and  Pins  in  Conic  Link-work. 

In  Fig.  322  is  given  a  photo-process  copy  of  an  example  of  conic 
link-work  for  equal  cone  slants,  and  where  the  angle  between  the 
axes  A  and  B  is  equal  to  that  be- 
tween the  pins  D  and  E  when  in 
one  plane  as  in  Fig.  320. 

Thus  conditioned,  the  cranks 
will  turn  continuously  in  the  same 
direction  as  in  Fig.  312,  or  in  op- 
posite directions,  as  in  Fig.  313. 

The  dead  center  accompanies 
conic  link-work,  and  gabs  and  pins 
may  be  used  here  as  well  as  in  the 
case  of  axes  parallel.  For  Fig.  322,  FIG.  322. 

the  elliptic  conic  pitch  lines  of  Prof.  MacCord  will  serve  to  deter- 
mine the  locations  of  the  gabs  and  pins,  those  curves  serving  as  the 
rolling-wheel. equivalents  of  this  linkage. 

Other  linkages  may  have  the  rolling-curve  equivalents  deter- 
mined as  in  Fig.  321,  when  the  location  of  the  gabs  and  pins  is 
readily  made. 

EXAMPLES   OF    PECULIAR   MOVEMENTS. 

Example  1. — In  Fig.  323  is  what  we  might  term  a  bent-shaft 
movement.  A  plunger  is  made  to  slide  in  the  top  head  in  a  direc- 
tion parallel  to  the  shaft. 

The  joint  work  could  be  simplified  by  using  a  block  and  two 
pins,  as  in  Fig.  324.  If  B  is  a  square  bar  the  center  line  of  the 
pin  D  should  strike  the  intersection  0. 

The  velocity-ratio  is  the  same  as  for  the  swash-plate  movement 
shown  in  Willis,  p.  172,  where  it  is  proved  that  the  motion  of  the 
bar  is  the  same  as  that  of  the  crank  and  crosshead,  with  infinite 
pitman. 

The  motion  is  still  the  same,  if  in  place  of  the  angular  part  of 
the  shaft  in  Fig.  323  an  enlarged  bearing  were  used,  like  an  ordi- 


282 


PRINCIPLES   OF   MECHANISM. 


nary  eccentric  and  strap,  and  mounted  on  the  straight  shaft  central, 
though  at  an  angle  like  that  of  the  swash  plate. 

A  movement  like  this  has  been  used  for  working  the  valve  in  a 
steam  engine  exhibited  at  the  Centennial  of  1876. 


FIG.  323. 

Example  2. — The  Hooke's  universal  joint  is  often  employed  as 
a  shaft  angle-coupling,  where  the  velocity-ratio  is  an  important 
consideration. 

The  joint  is  shown  in  Fig.  325,  where  A  and  B  are  the  shafts 
to  be  connected,  AD  and  FEE  half- 
hoops  between  which  is  a  cross  with 
one  branch  at  EF,  parallel  to  the 
paper,  and  the  other  at  D  perpen- 
dicular to  the  paper.  The  branches 
of  the  cross  are  pivoted  at  E  and  F9 
and  at  the  two  points  D,  in  AD. 

In  the  position  shown  the  velocity 
of  A  is  greatest,  and  the 

DP 

velocity-ratio  =  — =,, 

while  at  a  90-degree  turn  from  the  position  shown  the  velocity  of 
B  is  greatest  when  the 


Fm 


velocity-ratio  = 


II.    AXES    MEETING.  283 

because  for  the  first  position  A  acts  in  effect  by  an  arm  aF  upon 
an  arm  DF  from  B,  since  aFis  continually  in  the  plane  of  the  axis 
of  A,  and  of  the  arm  DF  of  the  cross. 

If  A  revolves  uniformly,  the  ratio  of  the  fastest  for  B  to  the 
slowest,  since  db  =  aF,  is 

Fastest  for  B  _  IDFV 
Slowest  for  B  ~~  \~aFj  ' 

These  limiting  speeds  are  in  the  same  ratio  as  those  for  elliptic 
wheels  of  Fig.  133,  where  DF  and  AF  are  the  distances  from  a 
focus  to  the  remote  and  a  nearer  vertex  respectively,  though  in  the 
latter  we  have  but  one  max.  and  one  min.  speed  in  a  revolution, 
while  in  Fig.  325  we  have  two.  But  the  law  of  variation  of  veloc- 
ity between  extremes  is  not  the  same  as  for  the  elliptic  wheels. 

In  the  case  of  a  pair  of  two-lobed  elliptic  wheels,  with  the 
maximum  and  minimum  diameters  in  the  same  ratio  as  DF  to  aF, 
the  ratio  of  the  limiting  speeds  is  the  same  as  for  Fig.  325,  and  the 
number  of  changes  in  a  revolution  is  the  same,  but  the  law  of  vari- 
ation of  velocity  still  remains  different. 

A  pair  of  overhead  shafts  A  and  B  connected  by  this  joint,  when 
the  angle  ADB  differs  much  from  180  degrees,  will  be  accompanied 
by  too  great  a  variation  of  velocity- 
ratio.  A 

In  this  case  the  use  of  two  joints 
between  A  and  B,  arranged  as  in  Fig. 
326,  with  the  branches  .Zi^and  HI 
of  the  crosses  in  the  same  plane 
and  with  the  angle  A  OB  =2  GDO, 
will  serve  to  transmit  motion  from 
A  to  B,  with  the  velocity-ratio  con- 
stant. 

For  the  single  joint  of  Fig.  325 
the  law  of  velocity-ratio  is  given  by  IG* 

the  formula  brought  out  in  connection  with  Fig.  256,  viz., 

ang.  velocity  of  A  __  1  —  sin2  x .  sin"  BDJ 
ang.  velocity  of  B  cos  BDJ 

in  which  x  is  the  angle  of  movement  of  A  from  the  positions  shown. 

This  equation  may  be  constructed  and  solved  graphically  by  the 

diagram,  Fig.  327.     Thus,  draw  BDJ  with  the  same  angle,  BDJ,. 


284 


PRINCIPLES   OF   MECHANISM. 


as  in  Fig.  325.     Lay  off  DJ  =  1  by  some  scale,  and  draw  JL  and 
L M  perpendicular  to  DL  and  DM  respectively. 


L    R       S      B 


Then 


and 


FIG.  327. 


JL  =  sin  BDJ 


JM  =  JL  sin  BDJ  =  sin'  BDJ. 


and 


Draw  the  semicircle  JPM  and  lay  off  any  angle  x  =  JMP. 
Then 

JM  sin  x  =  JP 
sin  x  =  JQ  =  JM  sin8  x. 


Then 

Z>()  =  DJ  -  JQ  =  1  -  /Jf  sin8  a?  =  1  -  sin8  g  .  sin8 
Draw  the  line  PS,  and  we  have 

DS  =  DQ  sec  £J9/  = 
Hence  the 
velocity-ratio  of  A  to  B  =  DS  =- 


cos  2?O/ 


cos  BDJ 


This  supposes  DJ  —  1,  but  if  it  equals  any  other  value,  measure 
.A/and  also  DS  by  the  same  scale,  and  divide  DS  by  ZVfor  the 
velocity-ratio. 


II.   AXES   MEETING.  285 

In  a  practical  case  draw  BDJ,  JL,  LM  and  the  semicircle  JPM, 
as  explained.  This  much  is  constant  for  a  particular  case  of  an 
angle  BDJ.  Then  lay  off  all  angles  x  the  velocity-ratio  is  desired 
for,  drawing  the  lines  MN9  MP,  etc.  Then  project  lines  NR9  PS, 
etc.,  perpendicular  to  DJ  when  we  have  the  series  of  velocity-ratios 
DR,  DS,etc. 

The  velocity-ratio  DL  is  a  minimum  and  BD  a  maximum;  the 
latter  answering  to  the  position  shown,  where  x  =  0. 

If  A  revolves  uniformly,  the  fastest  for  B  divided  by  the  slowest 
gives 

Fastest  for  B  _  DB  _  IDJ\*  _  (DF_V 
Slowest  for  B  ~~DL  ~  \DLJ        \aF)' 

since  from  Fig.  327 

DL  :DJ  I'.DJi  DB, 
or 

a  DB 

and 


Thus  the  results  of  Fig.  327  agree  with  that  obtained  from  Fig. 
325. 

This  example,  the  Hooke's  joint,  is  found  in  various  forms  of  con- 
struction, the  simplest  being  that  for  couplings  for  tumbling  rods 
for  transmitting  power  from  the  "horse-power"  to  the  threshing- 
machine  in  agricultural  districts.  On  each  end  of  each  rod  is  a 
forked  casting,  much  like  that  in  Fisr.  324,  with  bosses  through 
which  a  pin  may  be  placed  at  right  angles  to  the  rod.  A  block  goes 
loosely  between,  with  holes  at  right  angles  and  just  missing  each 
other.  Two  pins  or  bolts  are  used  at  each  joint,  each  passed  through 
a  fork  and  the  block,  and  at  right"  angles.  Thus,  between  adjacent 
rods  is  a  universal  joint,  so  that  the  series  of  rods  may  lie  upon  the 
ground  upon  notched  blocks,  or  on  blocks  and  between  stakes. 

For  a  considerable  angle  between  rods  at  a  joint  some  end  play 
will  occur,  when  the  holes  for  the  bolts,  as  above,  pass  beside  each 
other. 

Example  3.  —  The  Almond,  Reuleaux,  and  other  joints  or  coup- 
lings, are  more  compact  than  the  Hooke's,  so  that  they  may  very 
readily  be  enclosed  in  an  oil-tight  case,  to  facilitate  lubrication. 

In  Fig.  328  is  the  Almond  coupling,  serving  to  couple  a  pair  of 


286 


PRINCIPLES   OF   MECHANISM. 


axes  A  and  B  at  an  angle,  and  with  constant  velocity-ratio  equal 
to  1. 

At  D  is  a  fixed  shaft,  at  the  intersection  of,  and  perpendicular 
to  the  plane  of  the  axes  A  and  B.     On  this  shaft  a  sleeve  slides 


FIG.  328. 

from  which  project  two  arms  E,  E,  subtending  the  angle  ADB, 
and  upon  the  ends  of  each  of  which  arms  is  a  ball  to  work  in  a 
socket  F  in  the  swinging  piece  FH,  the  latter  being  in  two  pieces, 
held  together  as  one  by  a  bolt  ab,  and  pivoted  to  J. 

When  A  revolves,  each  ball  F  is  compelled  to  travel  in  a  curve 
on  the  cylinder  whose  axis  is  D,  one  curve  being  identical  with  the 
other.  Since  both  joined  pieces  FH  are  identical,  it  follows  that 
the  motion  of  B  is  a  copy  of  that  of  A,  and  hence  the  velocity-ratio 
equals  1. 

This  joint  or  coupling  will  connect  shafts  at  any  angle  ADB, 
provided  the  angle  between  the  arms  EE  is  made  the  same.  If  the 
shafts  A  and  B  are  in  one  straight  line,  the  coupling  will  reverse 
the  directions  of  motion. 

In  Fig.  329  is  a  somewhat  different  joint  or  coupling,  where  the 
shafts  A  and  B  have  fixed  cranks  with  long  crank  pins,  extending 
through  the  sockets  or  sleeves  F,  the  latter  having  right-angled  pin 
holes  passing  without  meeting,  through  which  are  pins  fixed  in 
the  arms  E  of  the  central  piece.  The  latter  slides  on  the  fixed 
shaft  or  stud  Z>,  placed  as  before  at  right  angles  to  A  and  B, 
and  at  their  point  of  intersection. 


II.    AXES    MEETING. 


287 


It  is  readily  seen  that  the  parts  D,  EE,  and  FF  compel  the 
crank  pin  B  to  maintain  the  same  relation  to  the  shaft  B  as  the 
crank  pin  of  A  does  to  the  shaft  A,  so  that  the  velocity-ratio  is 
constant. 

The  shafts  may  here  also  be  connected  at  any  angle  from  0  to 
180  degrees,  or  until  interference  of  cranks  occurs,  the  angle  be- 
tween the  branches  EE  being  made  the  same  as  that  between  the 
shafts  A  and  B. 

Besides  this,  the  shaft  B  may  be  placed  at  any  height  above  or 
below  A  by  simply  making  D  of  suitable  length  and  placing  the 
arms  E  connecting  with  B  the  same  amount  above  or  below  those 


connecting  with  A  as  the  one  shaft  is  above  or  below  the  other. 
Thus  by  making  D  open,  and  extended  into  sliding  and  revolving 
gudgeons  above  and  below,  we  meet  the  case  of  connecting  by  link- 
work  axes  which  cross  without  meeting,  and  with  a  velocity-ratio 
unity. 

The  Reuleaux  coupling  is  shown  in  Fig.  330,  where  A  and  B 
are  the  shafts  connected,  C  a  head  on  their  ends  with  holes  at  right 
angles  to  A  or  B  for  a  joint  pin,  the  latter  passing  the  forked  ends 
of  EF  at  F,  permitting  the  ends  E  to  swing  freely,  which  ends  E  are 
sleeve-like,  receiving  the  round  prongs  of  the  head  D,  thus  connect- 
ing EE  through  D.  The  ends  of  the  prongs  of  D  nearest  F  may 
have  a  nut  and  collar  to  prevent  D  from  being  thrown  outward  in 
revolving. 

This  coupling  or  joint,  so  put  up  as  to  have  the  angle  ^J^Fequal 
the  angle  FFB,  will  have  a  velocity-ratio  constant. 

This,  as  well  as  Figs.  327  and  328,  should  have  firm  bearings 


288  PRINCIPLES    OF    MECHANISM. 

for  A  and  B,  close  to  the  movement,  also  in  the  former  ones  for  /),, 
an  advisable  arrangement  being  an  iron  framework  especially  for 
them. 

One  important  practical  advantage  of  this  latter  over  the  former 


FIG.  330. 

two  couplings  is  the  fact  that  the  mutual  sliding  of  parts  is  simply 
from  rotation  upon  pins,  according  to  the  true  ideal  in  link-work, 
while  in  the  former  we  have  this  combined  with  a  disproportion- 
ately large  amount  of  end  sliding. 

At  F  the  branches  of  each  part  £JFm&y  be  made  unequal  and 
one  piece  EF  like  another,  so  that  the  bosses  of  one  piece  EF  will 
lie  beside  those  of  the  other,  uniting  at  F\  and  still  admitting  the 
four  sockets  E  to  all  lie  in  one  plane  EEEE. 

In  Fig.  329,  as  suggested  by  Prof.  E.  A.  Hitchcock,  we  have  a 
means  of  connecting  the  shafting  in  one  shop  room  above  or  below 
another  by  link- work. 

In  fact  we  may  say  any  number  of  shafts  parallel  to  one  an- 
other in  any  one  plane,  by  means  cf  one  shaft  D,  Fig.  329,  parallel 
to  the  plane,  crossing  all  the  shafts  at  any  angle. 

This  last  statement  corresponds  with  placing  the  shafts  A  and 
B,  Fig.  329,  with  others  in  parallel  in  any  plane,  but  at  a  distance 
from  one  another  along  the  line  of  D,  where  the  latter  consists  of 
a  shaft  free  to  slide  endwise  in  bearings  with  a  single-crank  arm- 
piece  E  at  each  shaft  A,  B,  etc.,  all  keyed  upon  the  shaft  D  in 
parallel,  D  being  near  the  plane  of  the  shafts  and  parallel  to  it. 
See  Fig.  333. 


CHAPTER   XXVI. 
LINK- WORK— (CONTINUED). 

III.    AXES  CKOSSING  WITHOUT  MEETING. 
Ratchet  and  Click  Movements. 

IN  Fig.  331  we  have  the  typical  case  of  projection  of  axes  which 
are  not  parallel  and  not  meeting,  where  A  and  B,  A'  and  E'  are  the 
axes,  Da  and  Eb  the  crank  or  lever  arms,  and  ab  the  connecting 
link.  At  a  and  b  some  sort  of  universal  joints  are  necessary,  as,  for 
instance,  the  ball-and-socket. 

According  to  Fig.  267  it  would  appear  that  the  velocity-ratio 
may  be  obtained  by  giving  the  parts  a  small  rotation,  so  that  the 


FIG.  331. 

end  d  of  the  common  perpendicular  dj  makes  the  movement  df, 
which,  projected  upon  the  connecting  rod,  gives  effor  its  endlong 
component  of  displacement;  also,  that  the  end  of  the  common  per- 
pendicular gk  makes  the  movement  gi,  which  projected  upon  the 
connecting  rod  gives  lii  for  the  endwise  component  of  displacement,, 
which  components  ef  and  hi  must  be  equal,  allowing  permanence 
of  length  of  connecting  rod,  so  that  the  velocity-ratio  will  equal  the 

289 


290 


PKINCIPLES   OF   MECHANISM. 


ratio  of  the  angles  djf  to  gki,  each  determined  as  the  angle  between 
two  meridian  planes  containing  the  radii  jd  and^/,  or  kg  and  ki. 

To  avoid  the  universal  joint  at  a  and  I,  Prof.  Willis  introduces 
a  second  intermediate  piece  between  a  and  #,  as  at  Fig.  332, 


A 


B 


Q 


FIG.  332. 

The  shafts  are  shown  as  not  parallel  and  not  meeting,  by  the 
lines  O'A',  P' B'  in  elevation,  and  in  plan  by  the  lines  0 A  and  PB. 

At  H  is  a  crank  for  A,  and  DE  a  connecting  rod  with  the  three 
axes  meeting  at  0.  At  /is  a  crank  and  GF&  connecting-rod,  with 
the  three  axes  meeting  at  P.  At  J  is  an  intermediate  piece,  with 
axes  QP  and  QO  to  connect  DE  and  GF. 

The  parts  thus  connected  may  be  moved  about  on  their  straight 
axes  when  0  remains  a  fixed  point  of  the  intersection  of  the  axes 
AO,  DO,  and  QO,  while  P  remains  a  fixed  point  for  the  axes  BP, 
GP,  and  QP.  Also,  Q  is  the  point  of  intersection  of  the  axes  of 
J.  In  use  these  parts  have  long  bearing  surfaces  without  cramping 
or  endlong  sliding. 

This  mechanism  becomes  somewhat  complicated  for  connecting 
axes  that  do  not  meet.  In  many  practical  cases  some  sliding  is  not 
objected  to,  since  the  sliding  facilitates  lubrication,  and  by  admit- 
ting sliding  and  rotary  motion  combined  we  often  greatly  simplify 
the  mechanism,  as  in  the  following: 

Examples. 

Example  1. — In  Fig.  333  we  have  the  case  of  axes  crossing  uut 
not  meeting,  and  yet  have  but  one  piece  intermediate  between  +he 
cranks  upon  the  axes  A  and  B,  that  piece  serving  as  a  link  of  in- 
finite length. 

The  crank  pins  are  parallel  to  the  axes  which  support  them. 


III.    AXES   CROSSING   WITHOUT   MEETING.  291 

As  the  axes  are  both  fixed  in  direction,  it  follows  that  the  angle 


n 


FIG.  333. 

between  the  pins  is  fixed,  and  that  a  block  with  long  bearing  holes 
at  the  same  angle  may  be  employed  to  connect  the  crank  pins. 

Continuous  motion  is  possible,  except  for  the  dead  centers,  and 
these  may  be  passed  by  use  of  gabs  and  pins  at  the 
points  determined  in  position,  as  by  the  equivalent 
non-circular  gear  pitch  lines. 

Example  2. — In  Fig.  334  is  given  an  example 
which  has  been  adopted  in  certain  machines,  with 
hundreds  of  them  in  successful  operation  for  some 
years  past.  The  axes  A  and  B  are  at  right  angles 
and  not  meeting. 

The  double  sleeve  D  has  holes  at  right  angles 
fitting  the  pins  closely.  It  turns  out  to  be  a 
simple,  compact,  and  efficient  movement,  the  piece 
D  serving  as  a  short  link. 

Example  3. — In  Fig.  335  is  an  example,  in 
practical  use  in  machines,  of  a  case  of  axes  A  and 
B  at  right  angles  and  not  meeting,  the  piece  D 
working  on  an  eccentric  A  as  driver,  upon  which 
it  slides  to  accommodate  it  to  the  arc  F  about  the 
fixed  axis  B,  while  D  slides  upon  E  to  admit  of 
the  lateral  movement  of  the  eccentric. 

In  practice,  the  longitudinal  sliding  of  the 
knuckle  piece  D,  combined  with  the  rotary,  is 
found  an  advantage,  as  favoring  distribution  of 
lubricant,  and  smoothness  of  surfaces  in  wear. 

Example  4. — Here  link-work,  much  like  that  of  Fig.  295,  drives 


FIG.  334. 


292 


PRINCIPLES   OF   MECHANISM. 


a  pin  F  in  a  curve  much  like  that  of  FE,  while  a  lever  pivoted  at 
B  carries  a  pin  G,  between  which  and  F  is  a  knuckle  piece  D, 
working  freely  on  both  pins. 


FIG.  335. 


FIG.  336. 


As  the  pins  are  always  at  right  angles,  the  piece  D  may  have 
long  bearing  holes  at  right  angles. 

If  we  take  A  of  Fig.  295  as  driver,  and  B,  Eig.  336,  as  fulcrum 
of  follower,  these  axes  A  and  B  cross  without  meeting.  This  is  in 
successful  operation  in  hundreds  of  tacking  machines. 

Ratchet  and  Click  Movements,  Axes  Variously  Related. 

These  movements  are  properly  classed  with  link-work,  some- 
times called  intermittent  link-work,  because  one  end  of  the  click  or 
pawl  is  usually  supported  on  a  pin,  while  the  other  in  action  rests 
in  a  notch,  which  serves  nearly  as  if  pinned  at  that  point. 

Example  1. — A  Running  Ratchet  of  simple  form  is  shown  in 
Fig.  337,  such  as  was  employed  in  the  old-fashioned  sawmills  to 
feed  along  the  carriage  and  log.  A  is  a  fixed  pin  about  which  AD 
swings,  causing  the  end  E  of  the  click  DE  to  move  forward  and 
back.  As  E  moves  forward  the  teeth  of  the  wheel  B  are  engaged 
and  moved,  while  on  the  return  E  draws  back  over  the  teeth,  the- 
detent  click  F  engaging  a  tooth  and  preventing  the  return  of  the 
wheel. 

Arranged  as  in  the  figure,  D  moves  in  the  arc  of  a  circle  which, 
together  with  the  movement  of  EB,  causes  ED  to  change  position 


OF  THE 

UNIVERSITY 

*2^4TJFORH\^ 
III.   AXES   CKOSSIKG   WITHOUTMEETlSlG. 


293 


relative  to  EB,  giving  greater  liability  of  E  to  slip  out  for  one  posi- 
tion than  for  others.  Also,  this  change  of  position,  or  swinging  of 
E  in  the  notch,  gives  cause  for  wear.  To  reduce  this,  the  pin  D 
should  be  so  placed  as  to  swing  in  a  normal  to  a  line  AB,  or  better, 


FIG.  337. 

a  circle  about  B.  An  approximation  to  this,  while  using  AD,  is 
obtained  by  locating  D  to  swing  from  H  to  /,  for  which  the  position 
of  DE  with  respect  to  EB  is  nearly  constant  for  forward  move- 
ment of  E. 

In  other  cases  D  is  mounted  on  a  pin  centered  at  B,  or  on  the 
axis  By  so  that  ED  is  fixed  relative  to  EB  when  in  action,  as  in 
the  case  of  Fig.  339. 

Example  2.— A  Running  Ratchet  for  Varied  Step  Movement  is 
shown  in  Fig.  338,  as  used  in  ther- 
mometer-plate graduating  machines 
over  thirty  years  ago.  At  B  is  a  ver- 
tical axis  about  which  the  slotted 
table  GH  swings.  The  slots  db,  cd, 
•etc.,  each  have  a  stud  that  can  be 
made  fast  at  any  point  in  the  slot. 
These  studs  are  slotted  to  receive  a 
flexible  ratchet  strip  DF,  and  are 
made  fast  in  studs  by  thumb  screws, 
as  shown  at  J.  A  click,  EE,  as  long 
as  the  slots  works  the  ratchet  strip, 
and  moves  the  table  HG,  notch  by 
notch,  between  each  of  which  move-  FIG.  338. 

ments  the  graduating  tool  cuts  a  mark  in  the  scale  being  gradu- 
ated, the  slide  supporting  the  scale,  tool,  etc.,  not  being  shown, 


294 


PRINCIPLES   OF   MECHANISM. 


the  scale  and  its  slide  being  moved  in  a  straight  line  proportional 
to  the  angular  displacement  of  HG. 

The  ratchet  strip  DF  is  shown  concentric  with  J9,  when  it  gives 
a  uniform  scale;  but  it  may  be  set  in  any  position,  as  dotted,  on  the 
table  HG,  when  a  correspondingly  varied  scale  results — as  finer  at 
one  end,  or  the  other,  or  in  the  middle,  or  at  both  ends. 

This  variability  is  required  to  meet  the  case  of  glass  thermome- 
ter tubes  which  cannot  be  made  of  uniform  calibre. 

Example  3. — A  Reversible  Running  Ratchet  is  shown  in  Fig. 
339,  where  the  click  A  is  kept  in  engagement  either  way  by  a 
spring  action  at  C.  A  central  notch  for  C  holds  the  click  out  of 
engagement  when  desired. 

A  variety  of  forms  of  A  are  in  use,  especially  in  feed  motions 
for  machinists'  tools. 

Example  4. — A  Reversible  Varied  Rate  Running  Ratchet  is 
shown  in  Fig.  340,  where  O  is  continually  reciprocating  about  B, 
carrying  the  pawls  or  clicks  E  and  F  forth  and  back  on  the  click 


C 


FIG.  339. 


FIG.  340, 


guard  D.  As  H  moves  D  to  the  one  side  or  the  other  a  click  en- 
gages one  or  several  teeth,  and  moves  B  little  or  much,  according 
to  extent  of  movement  of  D. 

When  D  moves  to  the  left  the  wheel  B  is  turned  right-handed 
by  the  click  E,  or  left-handed  by  the  click  F  if  D  is  thrown  to  the 
right.  The  more  D  is  displaced,  the  more  rapidly  will  B  be  moved 
by  the  click,  and  when  central,  B  is  stationary. 

This  movement  is  found  in  Snow's  Waterwheel  Governor. 


RATCHET   MOVEMENTS. 


295 


The  click  guard  D  has  had  frequent  application  with  good 
results. 

Example  5. — A  Continuous  Running  Ratchet  is  shown  in  Fig. 
341,  where  the  wheel  is  moved  the  same  way  for  each  movement  of 
the  handle  J  one  way  or  the  other.  Thus  the  holding  or  detaining 
click  of  Fig.  337  is  here  made  a  working  click.  The  clicks  may 
push  or  pull  in  action,  the  former  being  usually  preferred  for  con- 
siderations of  strength  and  shown  above  in  full  lines,  while  the 
latter  are  dotted  in.  The  pins  D  and  E  are  sometimes  placed  in  a 
line  perpendicular  to  AB,  but  the  figure  gives  that  relation  which 
causes  least  turning  of  the  ends  of  the  clicks  in  the  teeth,  and 
consequently  least  wear. 

The  teeth  may  be  internal  as  a  possibility,  but  are  usually  ex- 
ternal. 

Forms  of  Teeth. 

At  D,  Fig,  342,  the  normal  d  to  the  working  surface  a  of  a  tooth 


FIG.  341.  FIG.  342. 

may  go  inside  of  D,  when  the  click  will  be  secure.  But  if  it  goes  out- 
side of  D,  as  for  the  case  that  b  were  the  working  face  of  the  tooth, 
the  click  will  be  insecure,  or  very  likely  to  fly  out  of  engagement. 

Again,  if  Dd  is  excessive,  the  wheel  will  turn  back  appreciably 
before  coming  to  bearing  against  the  click  after  it  drops  in. 

The  click  may  have  a  shape  for  engagement  with  a  common 
gear  tooth,  as  at  F. 

At  G  several  clicks  may  be  placed  on  the  same  pin,  and  of  dif- 


296 


PRINCIPLES   OF   MECHANISM. 


ferent  lengths,  one  engaging  after  another  and  serving  in  effect  to 
divide  the  tooth  pitch  into  parts  answering  to  a  finer  toothed 
ratchet  wheel.  This  may  be  called  a  differential  Ratchet, 

At  E  is  a  stationary  ratchet  click  serving  to  hold  the  wheel 
stationary  against  moving  either  way. 

Friction  Ratchets. 

In  Fig.  343,  E  may  represent  a  round  rod  and  D  a  washer  easily 
sliding  over  E,  except  when  cramped,  as  by  the  pull  F<.  when  the 
greater  the  pull  the  greater  the  binding  or  grip 
upon  JEy  so  as  to  hold  effectuallyc  This  form  of 
ratchet  has  been  used  in  the  Brush  electric 
light  lamp. 

The  part  J?  may  be  a  square  rod  or  a  flange 
on  a  wide  piece,  D  fitting  properly. 

This  form  of  ratchet  always  works  satisfac- 
torily when  new,  but  in  practice  very  soon  be- 
comes untrustworthy,  and  is  usually  abandoned, 
especially  under  heavy  service. 
Shoe  pieces  will  extend  the  wearing  surfaces,  as  in  Fig.  344,  and 
they  have  been  used  with  fluted  seats,  as  shown  in  plan. 

These  work  admirably  for  a  much  longer  time  than  the  more 
limited  bearing  surfaces  of  Figs.  342  or  343,  but  finally  become 


D 


FIG.  343. 


--- 


FIG.  344. 


worn  in  places  where  most  used,  and  fail  to  work  as  expected. 

Example  1. — A  Continuous  Friction  Ratchet  is  shown  in  Fig. 
345,  which  was  once  used  on  a  rock-drilling  machine  for  feeding 
the  working  parts  along  as  the  drilling  progressed.  At  D  is  a 


HATCHET   MOVEMENTS. 


297 


flange  fast  to  the  stationary  framework,  while  J  is  a  part  oi  the 
sliding  carriage,  G  and  F  being  the  grip  clicks,  one  holding  in  the 


Fro.  345. 

opposite  sense  of  the  other,  H  is  a  clamp  under  slight  friction  grip, 
that  can  be  moved  towards  G  as  the  drilling  advances  and  a  tappet 
comes  to  press  upon  it.  That  moves  G,  and  in  turn  F  moves  by  the 
spring  /.  F  prevents  /  from  moving  up,  and  G  opposes  its  down- 
ward movement  until  H  acts  upon  it. 

This  apparatus  operated  most  admirably  for  a  few  days,  but  soon 
liad  to  be  abandoned,  as  most  ratchets  of  this  kind. 

Example  2.— Wire  Feed  Ratchets,  as  in  Fig.  346,  with  teeth  to 
grip  the  wire,  the  lower  ones  reciprocating  to- 
gether while  the  upper  ones  are  stationary, 
always  operate  with  satisfaction  for  a  few 
hours,  for  feeding  wire,  when  the  teeth  become 
worn  and  slip,  making  the  contrivance  a  failure. 
Example  3.— Running  Face  Ratchets  have 
the  teeth  upon  the  sides  of  the  wheel  instead  of 
edge,  as  shown  in  Fig.  347. 
In  this  example,  recently 
adopted  with  success  in 
practice,  the  click  E  is  a 
full  circle,  as  well  as  the 
having  the  same  number  of 


- 


FIG.  346. 

wheel   ./>,  both 
teeth,  viz.,  100. 

Ratchet  Gearing  is  made  in  great  variety — 
too  much  so  to  attempt  full  description  here.  FIG.  347. 

An  elaborate  enumeration  and  illustration  will  be  found   in  Reu- 
leaux's  Constructor,  translated  by  Suplee. 


Sx 


PART  V. 

CHAPTER  XXVII. 
REDUPLICATION. 

THIS  term  is  applied  to  such  movements  as  consist  of  circular 
i  pulleys  or  sheaves,  and  flexible  connectors 
passing  over  them,  usually  made  fast  at  one 
end,  mostly  ropes,  employed  to  multiply  the 
pull,  or  the  motion.  Blocks  and  tackles  of 
ships  are  included. 

In  Fig.  348  is  a  simple  example  where  the 
rope  is  made  fast  at  G,  passes  down  around 
the  pulley  F,  returns  and  passes  over  the 
pulley  E,  and  on  to  some  point  Z),  where  a 
pull  may  be  exerted  to  raise  a  weight  at  F,  as 
in  the  case  of  hay  unloaders,  stackers,  etc. 

When  the  ropes  from  F  up  are  parallel,  the 
velocity-ratio  is  constant;  but  when  the  rope 
diverges  or  converges  upward,  the  velocity-ratio  is  variable,  that  is, 
the  velocity  of  F  as  compared  with  that  of  D. 

In  this  case  D  must  move  twice  as  far  as  F,  and  the  pull  exerted 
at  F  will  be  twice  as  great  as  at  D. 

Again,  by  making  F  the  driver,  D  will  be  moved  twice  as  far  as 
F,  and  the  pull  exerted  at  D  will  be  one  half  that  at  F}  so  that  the 

velocity-ratio  =  2,  or  1/2. 

In  Fig.  349,  F  is  raised  faster  than  in  Fig.  348.     To  find  the 
velocitv-ratio,  draw  abed  with  lines  parallel  to  the  principal  figure. 

298 


REDUPLICATION. 


290 


Then  if  D  is  pulled  a  distance  cbd,  F  is  raised  a  height  ab,  be- 
cause ac  is  perpendicular  to  GF,  and  represents  the  movement  of 


G 


D 


a  point  of  GF  near  F,  when  cbd  is  removed.    Hence,  for  Fig.  349, 

velocity  of  D  _  cbd  _  2FH 
velocity  of  F  ~~  ab  "  EF  * 

Applying  this  to  the  case  of  Fig.  348,  we  will  find  EF  =  FH,  and 
the  velocity-ratio  reduces  to  2,  as  before 
stated.  One  make  .of  a  hay-unloader  is 
as  shown  in  Fig.  350,  where  the  velocity- 
ratio  is  3  to  1  and  constant,  when  the 
ropes  are  parallel  as  shown. 

In  the  Case  of  Ships'  Tackle,  where 
several  sheaves  are  placed  in  a  block  with 
the  ropes  parallel,  the  velocity-ratio  can 
readily  be  made  out  on  the  same  principle 
as  in  the  above  figures.  The  ropes  are 
usually  nearly  parallel  in  blocks  and 
tackle,  so  that  the  velocity-ratio  is  nearly 
constant. 

In  Elevators  operated  by  hydraulic 
power,  the  lifting  rope  passes  from  the  cage  up  over  a  pnlley  at  the 
top  of  the  shaft  or  hoisting  way,  and  down  to  a  series  of  sheaves, 
and  finally  made  fast  to  a  stationary  hitch  at  the  lower  end. 

If,  for  instance,  />,  Fig.  351,  represents  the  cage,  G  a  set  of 
stationary  sheaves,  F  a  set  of  sheaves  attached  to  the  hydraulic- 
power  rod  at  F,  then  when  ^7is  drawn  back  D  will  be  elevated. 

Suppose  the  rope  passes  from  the  hitch  near  G.  to  and  arotfnd 
F,  and  then  over  to  G,  up  to  E,  down  to  D,  thus  placing  two  ropes 
between  the  pulleys  F  and  G.  Then  when  F  is  drawn  back  10  feet 
D  will  be  elevated  20  feet,  the  multiplying  being  as  many  times  as 
the  ropes  between  G  and  F.  The  raising  of  D  will  be  80  feet  for 


FIG.  350. 


300 


PRINCIPLES  OF  MECHANISM. 


8  ropes  between  F  and  G,  and  for  a  draft  movement  of  10  feet  for 
F.    The  velocity-ratio  will  be  8  at  the  same  time. 

i v 


E 


I) 


FIG.  351. 


The  Weston  Differential  Block,  shown  in  Fig.  352,  is  a  simple 
device  for  one  so  powerful. 

An  endless  chain  passes  around  F,  then  up  and  around  E,  then 
down  in  a  loose  "  fall "  D9  then  around  G,  slightly  larger  than  E9 


Fi«.  352. 


and  down  to  F  again ;  G  and  E  being  in  one  piece. 

A  chain  is  used  instead  of  a  rope,  to  serve  as  a  hitch  at  EG, 
which  are  sprocketed  to  hold  to  the  chain.  Thus  there  is  a  hitch, 
in  some  measure  equivalent  to  the  rope  hitch  in  the  preceding 
cases,  so  that  the  device  may  be  classed  here. 


REDUPLICATION.  301 

/ 

Fig.  353  shows  that  when  D  is  pulled  a  distance  HG,  L  is 
raised  by  the  amount  GH,  while  K  is  lowered  at  the  same  time 
by  the  amount  J7,  and  F  is  raised  1/2  (GH  —  IJ),  so  that  the 
velocity-ratio,  as  between  D  and  F,  is 

velocity  of  D          2HG  2R 

velocity  of  F~  HG  -  IJ  ~  R  -  r 

if  R  and  r  are  the  radii  of  the  two  sizes  of  sheaves,  EG  and  EL 
If  R  =  5"  and  r  =  4",  then  the  velocity -ratio  equals  10. 


INDEX. 


PAGE 

Addendum,  dedendum,  or  root  line 93,  107,  148,  161,  175, 178 

Alternate  motions,  circular,  teeth  for,  limited 190 

,  unlimited 191 

,  limited  and  unlimited  mangle-wheels 83-86 

,  pitch  lines  for 83 

,  teeth  for 123 

Annular  wheels,  epicycloidal  teeth  for 141 

,  interference 141 

,  involute  teeth 144 

,  interference 144 

Approximate  gear  teeth 155-162 

Belt  gearing,  any  position  of  pulleys 247 

,  barrel  and  fusee  for  chronometers 240 

,  chain  and  sprocket  256 

,  cone  pulleys  for  lathes,  etc  247 

,  draw-bridge  equalizer 239 

,  for  treadle 243 

,  gas-meter  prover  equalizer 238 

,  law  of  motion  given  to  find  the  wheels 237 

,  non-circular,  for  rifling  machine 241 

,  pulley  with  high  center 246 

,  quarter  twist,  guide  pulley *  246 

,  retaining  belt  on  pulley 245 

,  rope  transmission 251-254 

,  spinning-mule  snail,  etc 241 

,  the  belt  and  circular  pulley 244 

,  velocity-ratio,  circular  and  non-circular 236 

Bevel  gearing,  circular,  epicycloidal,  involute 175 

,  circular,  pitch  lines  for 8 

,  intermittent  motion 82 

,  non-circular  pitch  lines 62-74 

,  teeth  for  non-circular  wheels 113 

Cams,  in  general , » . .  192 

,  by  co-ordinates 192 

intersection. 193-195 

303 


304  INDEX. 

PAGE 

Cams,  conical 198 

,  roller  for 216,  217 

,  cylindric,  straight  path 196- 

.curved     "     197 

,  diameter  constant 206,  213. 

,  breadth         "        207,209 

,  easement  for 208,  204 

,  headdle  cam 202 

,  inverse,  velocity-ratio 211> 

,  law  of  motion  defined 199-202- 

,  return  of  follower  by  gravity 211 

by  spring,  cam  groove,  positive 212. 

,  roller,  and  its  pin  for 214,  217 

,  for  spherical  cam , 217 

,  best  form 216> 

,  at  salient  points 215 

,  spherical 198 

,  tarrying  points 202 

,  thick  and  irregular  follower  extremity 215 

,  uniform  motion  for 202 

,  velocity-ratio 194 

,  with  flat-footed  follower 199,  205,  208 

several  followers 208 

Circular  rolling  wheels,  pitch  lines 7 

,  teeth  for 128 

Chain  and  sprocket  gearing 256 

Clearing  curve  and  filleting 149 

,  possible 149- 

Conical  cams ... 198 

link-work 279 

pulleys  in  belt  gearing 247 

roller  for  cams 217 

wheels  for  gears,  circular 8 

Conjugate  gear  teeth  for  circular  wheels 145-147 

non-circular  wheels 102 

,  Sang's  theory  for 168 

Couplings,  Almond , 286 

,  axes  not  meeting,  for 287 

,  Hooke's 283 

,  Oldham's 220-224 

,  Reuleaux's 288 

Cutters  for  gear  teeth,  epicycloidal  and  involute 167 

Cycle a 

Dead-points,  in  link- work 274 

conic  link-work 281 

Directional  relation 4 

,  constant,  variable. . . 4,  T 


INDEX.  305 

PAGE 

Elliptic  wheels,  pitch  lines  for 30 

,  interchangeable  multilobes 33-35 

Epicycloidal  curves,  peculiar  properties 129 

engine,  to  form  gear-tooth  curves 164 

gear  teeth,  .annular  wheels 140 

,  flanks  radial 130 

concave 132 

convex 132,  140 

,  for  inside  pin  geariug 136 

,  interchangeable  sets  133,  140 

,  interference  of  annular  gears 141 

,  least  crowding  and  friction ]  37 

,  line  of  action 151 

,  pin  annular  wheels 136,  137 

,  and  pin  teeth 134 

,  rack  and  pinion 137-139 

,  two-tooth  pinion 136,  137 

tooth  cutters 167 

Escapements,  anchor,  for  clocks,  dead-beat 226 

,  chronometer,  for  watches  and  chronometers 234 

,  cylinder,  for  watches 231 

,  duplex,      "        "       233 

,  gravity,  for  clocks,  dead-beat 229 

,  lever,  for  watches 282 

,  pin-wheel,  for  clocks,  dead-beat 228 

.recoil,  "       "        227 

,  power  escapements 225 

Friction  wheels,  pitch  lines  for 7 

in  cams 213 

,  relieved  by  roller 214 

ratchets 296 

Grant's  gear-cutting  engine 168,  170 

Hindley's  corset-shaped  worm 188 

Hooke's  universal  joint  or  coupling 283 

Hyperbolic  wheels,  pitch  lines 36 

Hyperboloids,  circular  rolling 10 

Intermittent  motions,  circular,  locking  arcs 13 

,  easements 13 

,  spurs  and  segments 13 

,  bevel  and  skew  bevel 189 

,  non-circular,  plane  and  bevel 81,  82 

,  rolling  spurs  for 120 

,  solid  easement  segments 122-125 

,  teeth,  spurs,  and  segments 116-127 

Involute  bevel  gear  teeth 175 


306  INDEX. 


PAGE 

Involute  gear  teeth,  cutters  for 167 

,  described  with  log-spiral 142 

on  drawing-board 155 

,  line  of  action  for 151 

Line  of  action  of  pressures  between  teeth ,. . .  151 

in  belt  gearing 237 

centers,  contact 5 

Link- work,  axes  crossing  without  meeting 289 

,  or  skew- bevel  link -work,  velocity-ratio 289 

with  sliding  blocks 291,  292 

of  Willis 290 

,  axes  parallel 259 

,  Corliss  valve  motion 260 

,  nailing  machine  motion  261 

,  needle  bar  motion 260 

,  conic,  or  solid,  velocity- ratio 279 

,  Almond's  286 

,  examples 282-288 

,  gab  and  pin,  and  dead  center 281 

,  for  axes  not  meeting 287 

,  Hooke's  joint,  and  velocity  for 282,  283 

,  rolling  wheel  equivalent  for . 280 

,  Reuleaux's 288 

,  dead  points  in 274 

,  gabs  and  pins  to  pass  ...   264,  275 

,  path  for 276 

,  path  and  velocity  of  various  points 261 

,  peculiar  features 259 

,  rolling  curve,  equivalent  for     264 

examples 264-273 

as  ellipses 271 

for  crank  and  pitman 267 

as  hyperbolas 270 

as  link  and  drag  link 268 

for  parabolas 272 

Sylvester's  kite 268 

unsym metrical,  example 273 

,  sliding  blocks  and  links  for 263,  291 

,  velocity-ratio  for 258 

Machine,  and  parts  of 1 

-made  gear  teeth 164 

gear-tooth  cutters 166,  167 

Mangle  wheel  and  rack,  pitch  lines  for 84-86 

,  teeth  for 127 

Mechanism,  elementary  combinations  of 1 

,  primary  and  secondary  trains  of 1 


INDEX.  307 

PAGE 

Mechanism,  train  or  trains  of  1 

,  table  of  elementary  combinations  of 2 

Motion,  uniform,  angular 3 

Names  and  terms  in  mechanism 3 

gearing 93 

Non-circular  wheels,  plane , 19 

,  equal  log-spiral,  segmental 20-24 

,  blocking  tendency 107 

,  in  general 44 

,  for  intermittent  motions 81 

,  internal  or  annular Ill 

,  involute,  teeth  for 101 

,  laws  of  motion  given,  to  find  the  wheels. . .  48-53 

,  limit  of  eccentricity 106 

,  log-spiral  wheels,  complete 24-29 

,  one  wheel  given,  to  find  its  mate 44-47 

,  rolling,  in  extreme  eccentricity 109 

,  five  special  forms ....  20 

,  solutions  of  practical  problems 54-61 

,  bevel,  drawn  direct  on  normal  sphere 73 

.example 72 

,  in  general,  solution 69 

,  five  special  forms 62-66 

,  teeth  for 112 

,  teeth  for  skew-bevel 114 

Normal  sphere,  drawing  wheels  directly  on 62-65,  71,  78 

Odontograph,  Grant's 161 

,  templet 157 

, Willis' 158 

Olivier  spiraloid  teeth  for  skew  bevels 182 

,  contact  between 183 

,  flat  faces  away  from  gorge 185 

,  interchangeability  of 182 

,  interference  of 183 

,  practical  at  gorge 187 

,  result  of  example 184 

Parabolic  wheels,  equal  and  similar 35,  272 

,  equivalent  link-work 272 

,  transformed 41 

,  with  gabs  and  pins 276 

Path  of  contact  in  gearing 149-150 

,  limited 151 

Period 3 

Pin  and  slit  movement,  inverse  cam  219,  220 

Pitch  lines..  7 


308  INDEX, 

PAGE 

Pitch  lines,  rolling  in  non-circular  wheels 109,  110 

,  circumferential 94,  148 

,  diametral 94,  148 

Point  of  contact 5,  7 

,  between  the  axes 7 

,  outside     "      "      ....    8 

Practical  considerations  in  circular  gearing 148-163 

Rack  and  pinion,  epicycloidal  teeth , 139 

,  involute  " 144 

Radius,  rod  to  carry  templet  odontograph. 157 

tooth  templet 154 

Ratchet  and  click,  movements 292 

,  form  of  teeth  and  click 295 

,  running 292 

,  continuous 295 

,  face  ratchets 297 

,  friction  ratchets 296 

,  wire  feed 297 

,  reversible 294 

,  varied  rate 294 

,  for  varied  steps 293 

Return  of  follower  in  cams,  by  gravity 211 

,  by  spring,  by  second  cam,  by  groove 212 

Reduplication,  velocity-ratio 298 

,  elevator  pulley  and  rope 300 

,  parallel  ropes 298,  299 

,  ropes  not  parallel 298 

,  Weston's  differential  block 300 

Revolution,  a  complete  turn 4 

Rolling  contact 5 

curve,  equivalent  for  link  work 264 

hyperboloids 10,  75 

Roller,  in  cam  movements 214 

,  proper  form  of 216 

,  pin  for 217 

Rope  transmission,  short  and  long  stretch 253 

,  for  haulage  lines 254 

Rotation,  a  partial  turn 4 

Bang's  theory  of  conjugating  gear-teeth 168 

Skew-bevel  wheels,  circular  pitch  surfaces 10-13 

,  error  in  early  statement  of 11 

,  non-circular  to  given  law 76 

,  pin  teeth , 188 

link-work 289 

Sliding  contact,  in  general,  velocity-ratio , 87-90 

and  rolling  contact,  in  one  model 90 


INDEX.  309 

PAGE 

Speeds,  geometrical  series  of  speeds  in  cone  pulleys 247 

Swasey's  gear-cutting  engine 168 

Teeth  of  gear  wheels,  conjugate 102,  145 

,  blocking  tendency '. 107,  152 

,  for  bevel  and  skew-bevel  non-circular 112-115 

intermittent  and  alternate  motions 116-127 

involute  non-circular 101 

,  individually  constructed 101 

,  generation  of  tooth  curves  for 92-105 

templets  for 95-101 

,  limited  inclination,  hooking 103-105 

,  names  and  rules 93 

-  ,  short,  in  eccentric  wheels 107,  152 

,  trachoidal 99 

Templets,  for  generating  gear  teeth,  form  and  size 95-100 

,  motion  templets 127 

,  odontograph 157 

,  path  templets 193 

,  tooth      "        154 

Tooth  profile,  epicycloidal,  for  circular  wheels 128,  152 

,involute,         "        "  "      143,155 

,  for  non-circular  wheels 98 

Transformed  wheels,  and  examples 38 

,unilobed 39-42 

,  bevel,  several  examples 66-68 

Velocity,  angular 3 

,  velocity-ratio 3 

,  in  rolling  contact 5 

•ratio  in  belt-gearing 237 

link-work 268,279,289 

reduplication 298 

sliding  contact 87 


SHORT-TITLE   CATALOGUE 

OF  THE 

PUBLICATIONS 

OF 

JOHN  WILEY   &    SONS, 

NEW    YORK, 

LONDON:    CHAPMAN   &   HALL,  LIMITED. 
ARRANGED  UNDER  SUBJECTS. 


Descriptive  circulars  sent  on  application. 

B'>oks  marked  with  an  asterisk  are  sold  at  net  prices  only. 

All  books  are  bound  in  cloth  unless  otherwise  stated. 


AGRICULTURE. 

CATTLE  FEEDING— DISEASES  OF  ANIMALS — GARDENING,  ETC. 

Armsby's  Manual  of  Cattle  Feeding 12mo,  $1  75 

Downing's  Fruit  and  Fruit  Trees 8vo,  5  00 

Kemp's  Landscape  Gardening 12mo,  2  50 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

Lloyd's  Science  of  Agriculture 8vo,  4  00 

London's  Gardening  for  Ladies.     (Downing.) 12mo,  150 

Steel's  Treatise  on  the  Diseases  of  the  Ox , 8vo,  6  00 

"      Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

Grotenfelt's  The  Principles  of  Modern  Dairy  Practice.     (Woll.) 

12mo,  2  00 

ARCHITECTURE. 

BUILDING — CARPENTRY— STAIRS,  ETC. 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  7  50 

Birkmire's  Architectural  Iron  and  Steel 8vo,  3  50 

Skeleton  Construction  in  Buildings 8vo,  3  00 


Birkmire's  Compound  Riveted  Girders 8vo,  $2  00 

American  Theatres— Planning  and  Construction. 8vo,  300 

Carpenter's  Heating  and  Ventilating  of  Buildings 8vo,  3  00 

Freitag's  Architectural  Engineering 8vo,  2  50 

Kidder's  Architect  and  Builder's  Pocket-book Morocco  flap,  4  00 

Hatfield's  American  House  Carpenter 8vo,  5  00 

Transverse  Strains 8vo,  5  00 

Monckton's  Stair  Building — Wood,  Iron,  and  Stone 4to,  4  00 

Gerhard's  Sanitary  House  Inspection 16mo,  1  00 

Downing  and  Wightwick's  Hints  to  Architects 8vo,  2  00 

Cottages 8vo,  2  50 

Holly's  Carpenter  and  Joiner 18mo,  75 

Worcester's  Small  Hospitals-  -Establishment  and  Maintenance, 
including  Atkinson's  Suggestions  for  Hospital  Archi- 
tecture  12ino,  125 

The  World's  Columbian  Exposition  of  1893 4to,  2  50 

ARMY,  NAVY,  Etc. 

MILITARY  ENGINEERING — ORDNANCE — PORT  CHARGES,  ETC. 

Oooke's  Naval  Ordnance 8vo,  $12  50 

Metcalfe's  Ordnance  and  Gunnery 12mo,  with  Atlas,  5  00 

Ingalls's  Handbook  of  Problems  in  Direct  Fire Svo,  4  00 

"  Ballistic  Tables Svo,  1  50 

BucknilFs  Submarine  Mines  and  Torpedoes Svo,  4  00 

Todd  and  Whall's  Practical  Seamanship Svo,  7  50 

Mahan's  Advanced  Guard 18mo,  1  50 

"  Permanent  Fortifications.  (Mercur.). Svo,  half  morocco,  750 

Wheeler's  Siege  Operations Svo,  2  00 

Woodhull's  Notes  on  Military  Hygiene 12mo,  morocco,  2  50 

Dietz's  Soldier's  First  Aid 12mo,  morocco,  1  25 

Young's  Simple  Elements  of  Navigation..  12m o,  morocco  flaps,  2  50 

Reed's  Signal  Service 50 

Phelps's  Practical  Marine  Surveying Svo,  2  50 

Very's  Navies  of  the  World Svo,  half  morocco,  3  50 

Bourne's  Screw  Propellers 4to,  5  00 

3 


Hunter's  Port  Charges 8vo,  half  morocco,  $13  00 

*  Dredge's  Modern  French  Artillery 4to,  half  morocco,  20  00 

Record  of  the  Transportation   Exhibits    Building, 

World's  Columbian  Exposition  of  1893.  .4to,  half  morocco,  15  00 

Mercur's  Elements  of  the  Art  of  War 8vo,  4  00 

"       Attack  of  Fortified  Places 12mo,  2  00 

Chase's  Screw  Propellers 8vo,  3  00 

Winthrop's  Abridgment  of  Military  Law 12mo,  2  50 

De  Brack's  Cavalry  Outpost  Duties.     (Carr.) 18mo,  morocco,  2  00 

Cronkhite's  Gunnery  for  Non-com.  Officers 18mo,  morocco,  2  00 

Dyer's  Light  Artillery 12mo,  3  00 

Sharpe's  Subsisting  Armies 18mo,  1  25 

"              "              "       4 18mo,  morocco,  150 

Powell's  Army  Officer's  Examiner 12mo,  4  00 

Hoff's  Naval  Tactics 8vo,  150 

BrufFs  Ordnance  and  Gunnery 8vo,  6  00 

ASSAYING. 

SMELTING— ORE  DRESSING— ALLOYS,  ETC. 

Turman's  Practical  Assaying 8vo,  3  00 

Wilson's  Cyanide  Processes 12mo,  1  50 

Fletcher's  Quant.  Assaying  with  the  Blowpipe..  12mo,  morocco,  1  50 

Ricketts's  Assaying  and  Assay  Schemes 8vo,  3  00 

*  Mitchell's  Practical  Assaying.     (Crookes.) 8vo,  10  00 

Thurston's  Alloys,  Brasses,  and  Bronzes 8vo,  2  50 

Jvunhardt's  Ore  Dressing 8vo,  1  50 

O'Driscoll's  Treatment  of  Gold  Ores , 8vo,  2  00 

ASTRONOMY. 

PRACTICAL,  THEORETICAL,  AND  DESCRIPTIVE. 

Michie  and  Harlow's  Practical  Astronomy 8vo,  3  00 

White's  Theoretical  and  Descriptive  Astronomy 12ino,  2  00 

Doolittle's  Practical  Astronomy 8vo,  4  00 

Craig's  Azimuth. 4to,  3  50 

Gore's  Elements  of  Geodesy 8vn,  2  50 

3 


BOTANY. 

GARDENING  FOB  LADIES,  ETC. 

Westerinaier's  General  Botany.     (Schneider.) 8vo,  $2  00 

Thome's  Structural  Botany 18mo,  2  25- 

Baldwin's  Orchids  of  New  England 8vo,  1  50 

London's  Gardening  for  Ladies.     (Downing.) 12mo,  1  50 

BRIDGES,  ROOFS,  Etc. 

CANTILEVER — HIGHWAY — SUSPENSION. 

Boiler's  Highway  Bridges Svo,  2  00  ' 

*  "      The  Thames  River  Bridge 4to,  paper,  500 

Burr's  Stresses  in  Bridges Svo,  3  50 

Merriman  &  Jacoby's  Text-book  of  Roofs  and  Bridges.     Part 

I.,  Stresses Svo,  250 

Merriman  &  Jacoby's  Text-book  of  Roofs  and  Bridges.     Part 

II.,  Graphic  Statics.. Svo,  2  50 

Merrimau  &  Jacoby's  Text-book  of  Roofs  and  Bridges.     Part 

III.,  Bridge  Design Svo,  5  00 

Merriman  &  Jacoby's  Text- book  of  Roofs  and  Bridges.     Part 

IY.,    Continuous,    Draw,    Cantilever,    Suspension,    and 

Arched  Bridges (In  preparation). 

Crehore's  Mechanics  of  the  Girder Svo,  5  00 

Du  Bois's  Strains  in  Framed  Structures 4to,  10  00 

Greene's  Roof  Trusses Svo,  1  25 

"     •  Bridge  Trusses Svo,  250 

"        Arches  in  Wood,  etc Svo,  250 

Waddell's  Iron  Highway  Bridges Svo,  4  00 

"Wood's  Construction  of  Bridges  and  Roofs Svo,  2  00 

Foster's  Wooden  Trestle  Bridges 4to,  5  00 

*  Morison's  The  Memphis  Bridge Oblong  4to,  10  00 

Johnson's  Modern  Framed  Structures 4to,  10  00 

CHEMISTRY. 

QUALITATIVE — QUANTITATIVE — ORGANIC — INORGANIC,  ETC. 

Fresenius's  Qualitative  Chemical  Analysis.    (Johnson.) Svo,      4  00 

"         Quantitative  Chemical  Analysis.    (Allen.) Svo,      6  00 

"  "  "  "  (Bolton.) Svo,      1  50 

4 


Crafts's  Qualitative  Analysis.     (Schaeffer.) 12mo,  $1  50 

Perkins's  Qualitative  Analysis 12mo,  1  00 

Thorpe's  Quantitative  Chemical  Analysis 18mo,  1  50 

Classen's  Analysis  by  Electrolysis.     (Herrick.) 8vo,  3  00 

Stockbridge's  Rocks  and  Soils '. 8vo,  2  50 

O'Brine's  Laboratory  Guide  to  Chemical  Analysis 8vo,  2  00 

Mixter's  Elementary  Text-book  of  Chemistry 12m o,  1  50 

Wulling's  Inorganic  Phar.  and  Med.  Chemistry 12mo,  2  00 

Mandel's  Bio-chemical  Laboratory 12mo,  1  50 

Austen's  Notes  for  Chemical  Students 12mo,  1  50 

Schimpf's  Volumetric  Analysis 12mo,  2  50 

Hammarsten's  Physiological  Chemistry  (Maudel.) 8vo,  4  00 

Miller's  Chemical  Physics 8vo,  2  00 

Pinner's  Organic  Chemistry.     (Austen.) 12mo,  1  50 

Kolbe's  Inorganic  Chemistry. 12mo,  1  50 

Ricketts  and   Russell's  Notes  on   Inorganic  Chemistry  (Non- 
metallic)  Oblong  8vo,  morocco,  75 

Drechsel's  Chemical  Reactions.    (Merrill.) 12mo,  1  25 

Adriance's  Laboratory  Calculations 12mo,  1  25 

Troilius's  Chemistry  of  Iron 8vo,  2  00 

Allen's  Tables  for  Iron  Analysis 8vo,  3  00 

Nichols's  Water  Supply  (Chemical  and  Sanitary) 8vo,  2  50 

Mason's        "           "     8vo,  500 

Spencer's  Sugar  Manufacturer's  Handbook .  12mo,  morocco  flaps,  2  00 

Wiechmann's  Sugar  Analysis 8vo,  2  50 

"            Chemical  Lecture  Notes..' 12mo,  300 

DRAWING. 
ELEMENTARY— GEOMETRICAL— TOPOGRAPHICAL. 

Hill's  Shades  and  Shadows  and  Perspective 8vo,  2  00 

Mahan's  Industrial  Drawing.    (Thompson.) 2  vols.,  8vo,  3  50 

MacCord's  Kinematics 8vo,  5  00 

Mechanical  Drawing 8vo,  400 

"         Descriptive  Geometry 8vo,  3  00 

Reed's  Topographical  Drawing.     (II.  A.) 4to,  5  00 

Smith's  Topographical  Drawing.     (Macmillan.) 8vo,  2  50 

Warren's  Free-hand  Drawing    12ino,  1  00 

5 


Warren's  Drafting  Instruments 12mo,  $1  25 

"  Projection  Drawing 12mo,  150 

"  Linear  Perspective .'12mo,  100 

Plane  Problems , . .  .12mo,  1  25 

"  Primary  Geometry 12mo,  75 

"  Descriptive  Geometry 2  vols. ,  8 vo,  3  50 

"  Problems  and  Theorems 8vo,  250- 

"  Machine  Construction c 2  vols. ,  8vo,  7  50 

"  Stereotomy— Stone  Cutting 8vo,  2  50 

"  Higher  Linear  Perspective 8vo,  3  50 

Shades  and  Shadows 8vo,  300 

Whelpley's  Letter  Engraving 12mo,  2  00 

ELECTRICITY  AND  MAGNETISM. 

ILLUMINATION — BATTERIES — PHYSICS. 

*  Dredge's  Electric  Illuminations. . .  .2  vols.,  4to,  half  morocco,  25  00 

.  ^{<,         Vol.  II 4to,  750 

Niaudet's  Electric  Batteries.     (Fishback. ) 12mo,  2  50 

Anthony  and  Brackett's  Text-book  of  Physics 8vo,  400 

Cosmic  Law  of  Thermal  Repulsion 18mo,  75 

Thurston's  Stationary  Steam  Engines  for  Electric  Lighting  Pur- 
poses  12mo,  1  50 

Michie's  Wave  Motion  Relating  to  Sound  and  Light, 8vo,  4  00; 

Barker's  Deep-sea  Soundings 8vo,  2  00' 

Holmau's  Precision  of  Measurements 8vo,  2  OO1 

Tillman's  Heat 8vo,  1  50 

Gilbert's  De-magnete.     (Mottelay.) 8vo,  2  50 

Benjamin's  Voltaic  Cell 8vo,  3  00 

Reagan's  Steam  and  Electrical  Locomotives 12mo  2  00 

ENGINEERING. 

CIVIL — MECHANICAL— SANITARY,  ETC. 

*  Trautwiue's  Cross-section Sheet,  25- 

*  "           Civil  Engineer's  Pocket-book.  ..12mo,  rnor.  flaps,  5  00 

*  "           Excavations  and  Embankments 8vo,  2  00 

*  "           Laying  Out  Curves 12mo,  morocco,  2  50 

Hudson's  Excavation  Tables.    Vol.  II. .                8vo,  1  00 


Searles's  Fieia  Engineering 12mo,  morocco  flaps,  $3  00 

"       Railroad  Spiral 12mo,  morocco  flaps,  1  50 

Godwin's  Railroad  Engineer's  Field-book.  12mo,  pocket-bk.  form,  2  50 

Butts's  Engineer's  Field-book 12mo,  morocco,  2  50 

Gore's  Elements  of  Goodesy 8vo,  2  50 

Wellington's  Location  of  Railways 8vo,  5  00 

*  Dredge's  Penn.  Railroad  Construction,  etc. . .  Folio,  half  mor.,  20  00 
Smith's  Cable  Tramways 4to,  2  50 

' '      Wire  Manufacture  and  Uses 4to>  3  00 

Mahan's  Civil  Engineering.     (Wood.) 8vo,  5  00 

Wheeler's  Civil  Engineering 8vo,  4  00 

Mosely's  Mechanical  Engineering.     (Mahan.) 8vo,  5  00 

Johnson's  Theory  and  Practice  of  Surveying 8vo,  4  00 

Stadia  Reduction  Diagram.  .Sheet,  22£  x  28£  inches,  50 

*  Drinker's  Tunnelling 4to,  half  morocco,  25  00 

Eissler's  Explosives — Nitroglycerine  and  Dynamite 8vo,  4  00 

Foster's  Wooden  Trestle  Bridges 4to,  5  00 

Ruff uer's  Non-tidal  Rivers 8vo,  1  25 

Greene's  Roof  Trusses 8vo,  1  25 

Bridge  Trusses 8vo,  250 

"      Arches  in  Wood,  etc 8vo,  250 

Church's  Mechanics  of  Engineering — Solids  and  Fluids 8vo,  6  00 

"       Notes  and  Examples  in  Mechanics 8vo,  2  00 

Howe's  Retaining  Walls  (New  Edition.) 12mo,  1  25 

Wegmann's  Construction  of  Masonry  Dams 4to,  5  00 

Thurston's  Materials  of  Construction 8vo,  5  00 

Baker's  Masonry  Construction 8vo,  5  00 

"       Surveying  Instruments 12mo,  300 

Warren's  Stereotomy— Stone  Cutting 8vo,  2  50 

Nichols's  Water  Supply  (Chemical  and  Sanitary) 8vo,  2  50 

Mason's        "           "              "            "          "        8vo,  500 

Gerhard's  Sanitary  House  Inspection 16mo,  1  00 

Kirkwood's  Lead  Pipe  for  Service  Pipe 8vo,  1  50 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

Howard's  Transition  Curve  Field-book.. . .  .12mo,  morocco  flap,  1  50 

Crandall's  The  Transition  Curve 12mo,  morocco,  1  50 

7 


Crandall's  Earthwork  Tables    8vo,  $1  50 

Pattern's  Civil  Engineering ,8vo,  7  50 

"       Foundations 8vo,  500 

Carpenter's  Experimental  Engineering 8vo,  6  00 

Webb's  Engineering  Instruments 12mo,  morocco,  1  00 

Black's  U.  S.  Public  Works 1'f,%1 4to,  5  00 

Merriman  and  Brook's  Handbook  for  Surveyors. . .  .12mo,  mor.,  2  00 

Merriman's  Retaining  Walls  and  Masonry  Dams 8vo,  2  00 

Geodetic  Surveying 8vo,  2  00 

Kiersted's  Sewage  Disposal 12mo,  1  25 

Siebert  and  Biggin's  Modern  Stone  Cutting  and  Masonry. .  .8vo,  1  50 

Kent's  Mechanical  Engineer's  Pocket-book 12mo,  morocco,  5  00 

HYDRAULICS. 

WATEK-WHEELS — WINDMILLS — SERVICE  PIPE — DRAINAGE,  ETC. 

Weisbach's  Hydraulics.     (Du  Bois. ) 8vo,  5  00 

Merriman's  Treatise  on  Hydraulics 8vo,  4  00 

Oanguillet&  Kutter'sFlow  of  Water.  (Hering&  Trailtwine.).8vo,  4  00 

Nichols's  Water  Supply  (Chemical  and  Sanitary) . .  .8vo,  2  50 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

Ferrel's  Treatise  on  the  Winds,  Cyclones,  and  Tornadoes. .  .8vo,  4  00 

Kirkwood's  Lead  Pipe  for  Service  Pipe 8vo,  1  50 

Ruffner's  Improvement  for  Non-tidal  Rivers 8vo,  1  25 

Wilson's  Irrigation  Engineering 8vo,  4  00 

Bovey's  Treatise  on  Hydraulics 8vo,  4  00 

Wegmann's  Water  Supply  of  the  City  of  New  York 4to,  10  00 

Hazeu's  Filtration  of  Public  Water  Supply 8vo,  2  00 

Mason's  Water  Supply — Chemical  and  Sanitary 8vo,  5  00 

Wood's  Theory  of  Turbines , 8vo,  2  50 

MANUFACTURES. 

ANILINE — BOILERS— EXPLOSIVES— IRON— SUGAR — WATCHES- 
WOOLLENS,  ETC. 

Metcalfe's  Cost  of  Manufactures 8vo,  5  00 

Metcalf 's  Steel  (Manual  for  Steel  Users) 12mo,  2  00 

Allen's  Tables  for  Iron  Analysis 8vo,  3  00 

8 


West's  American  Foundry  Practice 12mo,  $2  50 

"      Moulder's  Text-book     12mo,  250 

Spencer's  Sugar  Manufacturer's  Handbook. . .  .12ino,  inor.  flap,  2  00 

Wiechmann's  Sugar  Analysis 8vo,  2  50 

Beaumont's  Woollen  and  Worsted  Manufacture 12rno,  1  50 

*Reisig's  Guide  to  Piece  Dyeing 8vo,  25  00 

Eissler's  Explosives,  Nitroglycerine  and  Dynamite 8vo,  4  00 

Reimann's  Aniline  Colors.     (Crookes.) 8vo,  2  50 

Ford's  Boiler  Making  for  Boiler  Makers 18mo,  1  00 

Thurston's  Manual  of  Steam  Boilers 8vo,  5  00 

Booth's  Clock  and  Watch  Maker's  Manual 12mo,  2  00 

Holly's  Saw  Filing 18mo,  75 

Svedelius's  Handbook  for  Charcoal  Burners 12mo,  1  50 

The  Lathe  and  Its  Uses ...8vo,  600 

Woodbury's  Fire  Protection  of  Mills 8vo,  2  50 

Bolland's  The  Iron  Founder 12mo,  2  50 

"                              "        Supplement 12mo,  250 

"        Encyclopaedia  of  Founding  Terms 12mo,  3  00 

Bouvier's  Handbook  on  Oil  Painting 12mo,  2  00 

Steven's  House  Painting 18mo,  75 

MATERIALS  OF  ENGINEERING. 

STRENGTH — ELASTICITY — RESISTANCE,  ETC. 

Thurston's  Materials  of  Engineering 3  vols.,  8vo,  8  00 

Vol.  I.,  Non-metallic 8vo,  200 

Vol.  II.,  Iron  and  Steel 8vo,  3  50 

Vol.  III.,  Alloys,  Brasses,  and  Bronzes 8vo,  2  50 

Thurslou's  Materials  of  Construction 8vo,  5  00 

Duker's  Masonry  Construction.  UJiv^l. 8vo,  5  00 

Lanza's  Applied  Mechanics 8vo,  7  50 

"        Strength  of  Wooden  Columns 8vo,  paper,  50 

Wood's  Resistance  of  Materials Hm$$i>.  i^H'i' 8vo,  2  00 

Weyrauch's  Strength  of  Iron  and  Steel.    (Du  Bois.) 8vo,  1  50 

Burr's  Elasticity  and  Resistance  of  Materials 8vo,  5  00 

Merriinan's  Mechanics  of  Materials .y/it^l'.J.'*^;. 8vo,  4  00 

Ohurch's  Mechanic's  of  Engineering — Solids  and  Fluids 8vo,  6  00 

9 


Beardslee  and  Kent's  Strength  of  Wrought  Iron 8vo,  $1  5 

Hatfield's  Transverse  Strains 8vo,  5  OO 

Du  Bois's  Strains  in  Framed  Structures 4to,  10  00 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  00 

Bovey's  Strength  of  Materials 8vo,  7  50- 

Spaldiug's  Roads  and  Pavements 12mo,  2  00 

Rockwell's  Roads  and  Pavements  in  France 12mo,  1  25 

Byrne's  Highway  Construction 8vo,  5  00' 

Pattou's  Treatise  on  Foundations 8vo,  5  00' 

MATHEMATICS. 

CALCULUS — GEOMETRY— TRIGONOMETRY,  ETC. 

Rice  and  Johnson's  Differential  Calculus 8vo,  3  50 

Abridgment  of  Differential  Calculus.... 8vo,  150 
"                  Differential  and  Integral  Calculus, 

2  vols.  in  1,  12mo,  2  50' 

Johnson's  Integral  Calculus 12mo,  1  50 

"        Curve  Tracing 12mo,  1  00 

"        Differential  Eq.uations — Ordinary  and  Partial 8vo,  350- 

"        Least  Squares 12rno,  1  50 

Craig's  Linear  Differential  Equations 8vo,  5  00 

Merriman  and  Woodward's  Higher  Mathematics 8vo,  5  00' 

Bass's  Differential  Calculus 12mo, 

Halsted's  Synthetic  Geometry 8vo,  1  50' 

"       Elements  of  Geometry c..8vo,  175 

Chapman's  Theory  of  Equations 12mo,  1  50 

Merriman's  Method  of  Least  Squares 8vo,  2  00 

Comptou's  Logarithmic  Computations 12mo,  1  50 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo,  1  50 

Warren's  Primary  Geometry 12mo,  75 

Plane  Problems 12mo,  125 

"        Descriptive  Geometry 2  vols.,  8vo,  3  50 

"        Problems  and  Theorems 8vo,  2  50 

"        Higher  Linear  Perspective 8vo,  3  50 

"        Free-hand  Drawing l'2mo,  1  00 

"        Drafting  Instruments l~mo,  1  25- 


Warren's  Projection  Drawing 12mo,  $1  50 

Linear  Perspective 12mo,  100 

Plane  Problems 12mo,  125 

Searles's  Elements  of  Geometry. 8vo,  150 

Brigg's  Plane  Analytical  Geometry 12mo,  1  00 

Wood's  Co-ordinate  Geometry 8vo,  2  00 

"       Trigonometry 12mo,  1  00 

Maban's  Descriptive  Geometry  (Stone  Cutting) 8vo,  1  50 

Woolf  s  Descriptive  Geometry Royal  8vo,  3  00 

Ludlow's  Trigonometry  witb  Tables.     (Bass.) 8vo,  300 

Logaritbmic  and  Other  Tables.     (Bass.) 8vo,  2  00 

Baker's  Elliptic  Functions 8vo,  1  50 

Parker's  Quadrature  of  tbe  Circle 8vo,  2  50 

Totten's  Metrology 8vo,  2  50 

Ballard's  Pyramid  Problem 8vo,  1  50 

Barnard's  Pyramid  Problem 8vo,  1  50 

MECHANICS-MACHINERY. 

TEXT-BOOKS  AND  PRACTICAL  WORKS. 

Daua's  Elementary  Mechanics 12mo,  1  50 

Wood's                             "           12mo,  125 

"               "                 "           Supplement  and  Key 1  25 

Analytical  Mechanics 8vo,  300 

Michie's  Analytical  Mechanics 8vo,  4  00 

Merriman's  Mechanics  of  Materials 8vo,  4  00 

Church's  Mechanics  of  Engineering 8vo,  6  00 

"        Notes  and  Examples  in  Mechanics 8vo,  2  00 

Mosely's  Mechanical  Engineering.     (Mahan.) 8vo,  5  00 

Weisbach's   Mechanics    of  Engineering.    Vol.   III.,   Part  I., 

Sec.  I.     (Klein.) 8vo,  500 

Weisbach's  Mechanics    of  Engineering.     Vol.   III.,   Part  I. 

Sec.  II.     (Klein.) 8vo,  500 

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11 


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